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THE 


NORMAL 

MENTAL  ARITHMETIC1 

A 

THOROUGH  AND  COMPLETE  COURSE, 


BY 


ANALYSIS  AND   INDUCTION. 

BY 

EDWAKD  BKOOKS,  A.M., 

PRINCIPAL  AND  PROFESSOR  OP  MATHEMATICS  IN  PENNSYLVANIA  STATE 
NORMAL  SCHOOL,  AND  AUTHOR  OF  "  NORMAL  PRIMARY  ARITHMETIC," 

"NORMAL  ELEMENTARY  ARITHMETIC,"  "  NORMAL  WRITTEN 
ARITHMETIC,"  "NORMAL  GEOMETRY,"  ETC. 
? 
OFTHE 

UNIVERSITY      &Kstb  Cbifio* 

WITHA 

TREATISE  ON  MENTAL  ALGEBRA 


•  A»*»yiis  and  Induction  are  the  golden  keys  which  unloeK  the  various  enmpies  comt 
nations  ot  numbcre,. " 


PHILADELPHIA : 
SOWER,     POTTS    &    CO., 

530  MARKET  ST.,  AND  523  MINOR  ST. 


OFFICE  OF  THE  CONTROLLERS  OF  PUBLIC  SCHOOLS,  ") 

FIRST  DISTRICT  OF  PENNSYLVANIA.  \. 

PHILADELPHIA,  March  29, 1869.    j 

At  a  meeting  of  the  Controllers  of  Public  Schools,  First  Dis- 
trict of  Pennsylvania,  held  at  the  Controllers'  Chamber,  Tues- 
day, March  9,  1869,  the  following  resolution  was  adopted:-— 

Resolved,  That  the  Normal  Series  of  Arithmetics,  comprising 
"Brooks's  Normal  Primary  Arithmetic,"  "Brooks's  Normal 
Elementary  Arithmetic,"  "Brooks's  Normal  Mental  Arithme- 
tic," and  ''Brooks's  Normal  Written  Arithmetic,"  by  Edward 
Brooks,  Esq.,  be  and  the  same  are  hereby  placed  on  the  list  of 
books  to  be  used  in  the  Public  Schools  of  this  District. 
From  the  minutes : 

H.  W.  HALLIWELL,  Sec'y. 


Entered,  according  to  Act  of  Congress,  in  the  year  1858,  by 
EDWARD  BROOKS, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  in  and  for  the 
Eastern  District  of  Pennsylvania. 

Entered,  according  to  Act  of  Congress,  in  the  year  1863,  by 

EDWARD  BROOKS, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  in  and  for  th« 
Eastern  District  of  Pennsylvania. 

MEARS  ft  DUSENBERY,  STEREOTYPERS.  SHERMAN  &  CO.,  PRINTERS. 


The  Board  of  School  Trustees  for  the  STATE  OP  MARYLAND, 
recommend  Brooks's  Normal  Arithmetics  and  Fewsmith's  Gram- 
mars, for  use  in  all  the  Public  Schools  of  that  state. 


The  Board  of  School  Commissioners  for  the  CITY  OP  BALTI- 
MORE, have  adopted  Brooks's  Normal  Series  and  Fewsmith'i 
Grammars,  for  exclusive  use  in  the  Public  Schools  of  that  city. 


10 


J     P.    WICKER  SH  AM,    A.  M 

PRINCIPAL  OF  THE  LANCASTER  COUNTY  NORMAL  SCHOOL, 


A  TOKEN  OF  HIGH  PERSONAL  "REGARD, 

AND 
.  TRIBUTE  OF  ADMIRATION  FOR  HIS  NOBLE  EFFORTS 

IN  THE 
GREAT  CAUSE  OF  POPULAR  EDUCATION, 

)p0  Jittte  fflotame 

18  M081  SINCERELY  INSCRIBED  BY 

THE  AUTHUS, 


165026 


PREFACE. 


THE  science  of  Arithmetic,  as  taught  unui  quite  re- 
cently, has  been  much  less  beneficial  as  an  educational 
agency  than  it  should  have  been.  Consisting  mainly 
of  dictated  methods  of  deriving  results,  without  present- 
ing the  reasons  for  the  various  operations,  it  failed  to 
afford  that  high  degree  of  mental  discipline,  which,  when 
properly  taught,  it  is  so  well  calculated  to  impart.  But 
a  great  change  has  been  wrought  in  this  respectn— a  new 
era  has  dawned  upon  the  world  of  science — the  royal  road 
to  Mathematics  has  been  so  graded  and  strewn  with  the 
flowers  of  reason  and  philosophy,  that  it  is  now  full  of 
interest  and  profit  to  the  youthful  learner ; — and  one  of 
the  most  influential  agents  in  this  work  has  been  the 
system  of  Mental  Arithmetic. 

The  present  is  a  proud  period  in  the  history  of  popular 
education.  At  no  previous  time  has  there  been  so  deep 
and  universal  an  interest  manifested  in  the  advancement 
of  the  cause ;  and  this,  combined  with  the  elevation  of 
the  standard  of  qualification  among  our  teachers,  has 
created  a  demand  for  a  more  philosophic,  systematic, 

(3) 


IV  PREFACE. 

and  comprehensive  text  book  on  the  subject  of  Mental 
Arithmetic,  than  has  hitherto  been  presented ; — and,  at 
the  solicitation  of  his  friends,  the  author  of  the  present 
work  has  attempted  to  meet  that  demand ;  how  success- 
fully, must  be  determined  by  those  whose  province  it  is  to 
decide. 

In  the  prosecution  of  this  object  he  has  availed  himself 
of  the  experience  of  several  years  of  actual  instruction  in 
the  science,  and  of  the  careful  examination  of  what  has 
been  previously  written,  for  the  purpose  of  ascertaining 
wherein  improvements  might  be  made.  He  does  not 
assume  that  every  problem,  or  class  of  problems,  or  the 
principle  of  every  solution,  is  original  with  himself;  as 
every  intelligent  teacher  knows  that  certain  problems  and 
solutions  have  stood  the  test  of  years,  and  no  book  would 
be  faultless  without  them.  The  object  has  been  not  to  be 
merely  original,  but  to  prepare  a  suitable  and  valuable 
text  book;  he  feels  confident,  however,  that  it  will  be 
found  to  contain  sufficient  distinct  and  original  features 
to  recommend  it  to  the  favourable  consideration  of  the 
friends  of  education. 

The  work,  like  others  of  the  same  class,  is  not  designed 
for  the  child's  first  book  in  the  science  of  numbers,  and, 
therefore,  the  more  elementary  operations  have  not  been 
uselessly  enlarged  upon ;  yet  the  arrangement  is  so  syste- 
matic, and  the  transition  from  the  easy  to  the  complex  so 
gradual,  that  even  very  young  pupils  can  pursue  it  with 
ease  and  advantage. 

A  system  of  Mental  Arithmetic  should  be  based  upon 
the  principles  of  Analysis  and  Induction.  Results  should 
be  derived  by  analytic  processes,  and  methods  inferred 
from  these  processes — and  this  is  the  philosophy  upon 


PREFACE  V 

which  the  present  treatise  is  founded.  Analysis  and  In- 
duction are  the  golden  keys  which  unlock  the  various 
complex  combinations  of  numbers — the  magic  wands  by 
which  the  intricate  and  abstruse  are  unfolded  in  logical 
simplicity.  The  great  element  of  Analysis  is  comparison, 
and  the  equation  is  the  great  Archimedean  lever  of  com- 
parison. It  enters  into  every  operation,  from  the  simplest 
combination  of  Arithmetic,  to  the  most  complicated  prob- 
lem of  the  transcendental  Analysis.  In  Geometry  the 
axioms  and  definitions  are  the  standards  of  comparison  j 
in  Algebra  we  compare  the  unknown  with  the  known  to 
determine  its  value ;  and  in  Arithmetic  we  compare  all 
numbers,  and  the  effects  produced  by  a  number  of  equal 
causes,  with  the  unit,  or  effect  of  the  single  cause.  And 
thus  the  science  of  Mathematics  is  evolved,  comprising 
a  vast  series  of  dependent  truths,  derived  from  successive 
comparisons  of  the  unknown  with  the  known,  the  theo- 
retic with  the  axiomatic,  the  complex  with  the  simple. 

In  the  science  of  numbers,  this  relation  of  the  col- 
lection to  the  unit  is  so  evident,  that  it  is  intuitively 
apprehended,  and  hence  the  simplicity  of  this  elementary 
process  of  Analysis.  But  as  the  pupil  progresses  in  the 
science,  it  will  be  perceived  that  different  collections  bear 
certain  relations  to  each  other,  and  he  should  be  taught 
to  discover  and  apply  these  new  relations.  To  develope 
this  theory  fully  and  completely  has  been  the  object  of 
the  author. 

Attention  is  also  called  to  the  arrangement  and  treat- 
ment of  Fractions.  First  is  given  the  fractional  word, 
treating  them  like  denominate  numbers,  then  the  nume- 
rical fractional  expression,  and  then,  after  extensive 
exercise  iu  analytic  processes,  mechanical  methods  are 


VI  PREFACE. 

derived  from  these  processes  by  Induction,.  At  the 
close  of  Fractions,  some  of  these  methods  are  stated  ID 
the  form  of  Propositions,  and  their  truth  substantiated 
by  demonstration,  in  a  manner  different  from  any  which 
the  author  has  met  with,  and  which  he  has  employed  in 
teaching,  with  entire  satisfaction.  In  nearly  every  sectioD 
of  the  book  will  be  found  matter  not  previously  introduced 
into  works  of  this  kind,  and  much  that  has  never  before 
been  published.  Questions  upon  definitions  and  principles 
are  given  in  various  parts  of  the  work,  without  answers  to 
them,  the  object  being  to  awaken  original  thought  with 
the  pupil,  which  is  deemed  far  more  valuable  than  the 
most  accurate  definition. 

But  it  is  impossible,  as  well  as  unnecessary,  to  call  the 
attention  to. all  the  different  peculiarities  of  the  work, 
since  intelligent  teachers,  upon  examination,  will  see  and 
judge  for  themselves.  The  whole,  with  its  merits  and 
demerits,  is  respectfully  submitted  to  the  public,  with 
the  sincere  wish  that  it  may  prove  a  valuable  auxiliary 
in  the  promotion  of  that  noble  science,  which  promises 
BO  much  for  the  advancement  of  the  great  cause  of  popu- 
lar education. 

E.  B. 

LANCASTER  COUNTY  NORMAL  SCHOOL, 
April  7,  1858, 


SUGGESTIONS  TO  TEACHERS. 

THE  attention  of  teachers  is  respectfully  solicited  to  the  following  Method 
of  Recitation.  Some  of  them  are  preferable  to  others,  but  all  may  occasionally 
bo  used  with  advantage. 

COMMON  METHOD.— By  this  method  the  problems  are  read  by  the  teacher  and 
assigned  promiscuously,  the  pupils  not  being  permitted  to  use  the  book  during 
recitation,  nor  retain  the  conditions  of  the  problems  by  means  of  pencil  and 
paper,  as  is  sometimes  done.  The  pupil  selected  by  the  teacher  arises,  re- 
peats the  problem,  and  gives  the  solution,  at  the  close  of  which  the  mistake? 
that  may  have  been  made  should  be  corrected  by  the  class  or  teacher. 

SILENT  METHOD.— By  this  method  the  teacher  reads  a  problem  to  the  class, 
and  then  the  pupils  silently  solve  it,  indicating  the  eompletion  of  the  solution 
by  the  upraised  hand.  After  the  whole  class,  or  nearly  the  whole  class,  have 
finished  the  solution,  the  teacher  calls  upon  some  member,  who  arises,  repeats 
the  problem,  and  gives  the  solution  as  in  the  form<?r  m«uhod. 

By  this  method  the  whole  class  must  be  exercised,  upon  every  problem,  thus 
securing  more  discipline  than  by  the  preceding  method.  It,  however,  requires 
more  time  than  the  first;  hence,  not  so  many  problems  can  be  solved  at  a  reci- 
tation. We  prefer  the  first  method  for  advanced  pupils,  and  the  second,  at 
least  a  portion  of  the  time,  with  younger  pupils. 

CHANCE  ASSIGNMENT. — This  method  differs  from  the  first  only  in  the  assign- 
ment of  the  problems.  The  teacher  marks  the  number  of  the  lesson  and  the 
number  of  the  problem,  upon  small  pieces  of  paper,  which  the  pupils  may  take 
out  of  a  box  passed  around  by  the  teacher  or  some  member  of  the  class.  The 
teacher  then,  after  reading  a  problem,  instead  of  calling  upon  a  pupil,  merely 
gives  the  number  of  the  problem,  the  person  having  the  number  arising,  repeat- 
ing, and  solving  it.  By  this  method  the  teacher  is  relieved  of  all  responsibility 
with  reference  to  hard  and  easy  problems,  and  it  is  also  believed  that  better 
attention  is  secured  with  it.  It  is  particularly  adapted  to  reviews  and  public 
examinations. 

DOUBLE  ASSIGNMENT. — By  this  method  the  pupil  who  receives  the  problem 
from  the  teacher,  arises,  repeats  it,  and  then  assigns  it  to  some  one  else  to 
solve.  It  may  be  combined  with  cither  the  first  or  second  methods.  The 
objects  of  this  method  are  variety  and  interest. 

METHOD  BY  PARTS. — By  this  method  different  parts  of  the  same  problem  are 
solved  by  different  pupils.  The  teacher  reads  the  problem  and  assigns  it  to  a 
pupil,  and  after  he  has  given  a  portion  of  the  solution,  another  is  called  upon, 
who  takes  up  the  solution  at  the  point  where  the  first  stops ;  the  second  is  suo 
oeeded  in  like  manner  by  a  third ;  and  so  on,  until  the  solution  is  completed. 
The  object  of  this  method  is  to  secure  the  attention  of  the  whole  class,  which  it 
does  very  effectually.  It  is  particularly  suited  TO  a  large  class  consisting  of 
young  pupils. 

UNNAMED  METHOD.— By  this  method  the  teacher  reads  and  assigns  several 
problems  to  different  members  of  the  class,  before  requiring  any  solutions,  after 
which  those  \*bo  have  received  problems  are  called  upon  in  the  order  of  assign- 

(vii) 


Vlii  SUGGESTIONS   TO    TEACHERS. 

ment  fin  Iheir  solutions.  The  advantages  of  this  method  are,  first,  the  pupil, 
having  some  time  to  think  of  the  problem,  is  enabled  to  give  the  solution  with 
more  promptness  and  accuracy,  and  secondly,  the  necessity  of  retaining  th« 
numbers  and  their  relations  in  the  mind  for  several  minutes,  affords  a  good 
discipline  to  the  memory. 

CHOOSING  SIDES. — This  is  a  modification  of  the  old  spelling  class  method,  and 
Is  one  calculated  to  elicit  a  very  great  degree  of  interest.  By  it  two  pupils,  ap 
pointed  by  the  teacher,  select  the  others,  thus  forming  two  parties  for  a  trial 
of  skill,  as  in  a  game  of  cricket  or  base  ball.  The  problems  may  be  assigned 
alternately  to  the  sides,  by  the  teacher,  by  chance,  by  the  leaders  of  the  sides, 
or  in  any  other  way  that  may  be  agreed  upon  by  the  teacher  and  class. 

In  regard  to  these  methods,  the  first,  second,  and  third  are  probably  the  best 
for  the  usual  recitations,,  but  the  other  methods  can  very  profitably  be  employed 
with  younger  classes,  or,  in  fact,  with  any  class,  to  relieve  monotony  and  awaken 
interest.  With  advanced  pupils  we  prefer  the  first  method,  or  the  first  com- 
bined with  the  third. 

ERRORS  TO  BE  AVOIDED. 

There  is  a  large  number  of  errors  to  which  pupils  in  every  section  of  the 
country  are  liable,  a  few  of  which  we  will  mention.  We  classify  them  as  errors 
of  Position  and  errors  of  Expression. 

ERRORS  OF  POSITION. — Pupils  are  exceedingly  liable  to  assume  improper  posi- 
tions and  awkward  attitudes  during  recitation,  such  as  leaning  on  the  desk  or 
against  the  wall,  putting  the  foot  upon  a  seat,  jamming  the  hands  in  the  pockets, 
particularly  when  the  problem  is  bard,  playing  with  a  button,  watch-chain,  &c. 
All  of  these  laults  should  be  carefully  guarded  against,  for  reasons  so  obvious 
that  they  need  not  be  mentioned.  An  erect  and  graceful  carriage,  aside  from 
its  relation  to  health,  is  of  advantage  to  every  lady  and  gentleman. 

ERRORS  OF  EXPRESSION. — Under  this  head  we  include  errors  of  Articulation, 
Pronunciation.  Grammar,  &c.  There  is  quite  a  large  number  of  words  which 
pupils  in  their  haste  mispronounce,  and  also  quite  a  large  number  of  combina- 
tions, which  by  a  careless  enunciation  make  ridiculous  sense,  or  nonsense.  We 
will  call  the  attention  to  a  few  of  them,  suggesting  to  the  teacher  to  correct 
these  and  others  he  may  notice. 

"  And"  is  often  called  "  an ;"  "/or"  is  called  "fur ;"  "  o/"  is  pronounced  as 
If  the  o  was  omitted ;  words  commencing  with  wh,  as  when,  which,  where,  &c., 
are  pronounced  as  if  spelled  "  wen,"  "  wich,"  "  were"  &c. 

"Gave  him"  is  called  "gavim;"  "did  he"  is  called  "diddy;"  "had  he"  is 
called  '•  haddy  ;"  "give,  him"  is  called  "givim;"  "give  ?ier"  is  called  "giver;" 
" which  is"  is  often  changed  into  "witches;"  and  "how  many"  is  frequently 
transformed  into  "hominy"  "How  many  did  each  earn"  is  often  rendered 
*  hominy  did  e  churn." 

A  very  common  error,  and  one  exceedingly  difficult  to  correct,  is  involved  in 
the  following  solution:  "If  2,  apples  cost  6  cents,  one  apple  will  cost  the  i  of  6 
cents,  which  are  3  cents."  Here  "  the"  is  superfluous,  and  "  are"  is  ungram- 
matical. 

The  following  is  a  frequent  error:  "If  one  apple  cost  3  cents,  for  12  cents  you 
can  buy  as  many  apptes  as  3  is  contained  in  12,  which  are  4  times"  The  objec 
tions  are,  first.  3  is  not  contained  any  apples  in  12;  secondly,  the  result  obtained 
is  times,  when  it  should  be  apples,  or  a  number  which  applies  to  both  times  and 
apples.  The  solution  should  be,  "  You  can  ouy  as  many  apples  for  12  cents  ca  3 
is  contained  times  in  12,  which  are  4." 


MENTAL    ARITHMETIC. 


SECTION    I. 

LESSON  L 

1.  If  I  have  2  cents  in  one  hand,  and  1  cent  in  the 
other,  how  many  cents  have  I  in  both  ?' 

SOLUTION.  —  If  I  have  2  cents  in  one  hand,  and  1  cent  in  the 
other,  in  both  hands  I  have  2  cents  plus  1  cent,  which  are  3  cents. 
Therefore,  if  I  have  2  cents  in  one  hand,  and  1  cent  in  the  other, 
I  have  3  cents  in  both. 

2.  John  has  3  apples,  and  William  has  2  \  how  many 
have  they  hoth  ? 

3.  Fanny  has  4  peaches,  and  her  sister  has  3  ;  how 
many  have  they  both  ? 

-    ,4-  James  is  3  years  old,  and  Sarah  is  6  ;  what  is  the 
sum  of  their  ages  ? 

5.  Joseph  bought  6  peaches,  and  his  brother  gave  him 
4  ]  how  many  had  he  then  ? 

6.  I  paid  10  cents  for  a  slate,  and  2  cents  for  a  pencil  j 
i  what  did  they  both  cost  ? 

7.  A  cap  cost  3  dollars,  and  a  coat,  12  dollars  ;  what 
did  they  both  cost  ? 

8.  Peter  had  8  cents,  and  found  10  more;  how  many 
had  he  then? 

9.  Martin  had  6  birds,  and  caught  9  more  ;  how  many 
had  he  then  ? 

(7) 


8  MENTAL  ARITHMETIC. 

10.  A  pig  cost  5  dollars,  and  a  sheep,  7  dollars ;  what 
did  they  both  cost  ? 

11.  If  Cyrus  had  9  cows,  and  bought  11  more,  how 
many  would  he  then  have  ? 

12    A  saddle  cost  8  dollars,  and  a  harness,  13  dollars  5 
how  much  did  they  both  cost  ? 

13.  Sally  had  10  pins  in  her  cushion,  and  put  in  9 
more ;  how  many  were  then  in  the  cushion  ? 

14.  If  James  rode  8  miles,  and  walked  13,  how  far  did 
e  travel  ? 

15.  Mustard  has  10  sisters  and  11  brothers ;  how  many 
are  there  in  the  family  ? 

16.  Taylor  had  5  horses,  and  his  brother  sold  him  13  \ 
how  many  had  he  then  ? 

17.  In  a  garden  there  are  9  plum  trees,  and  11  peach 
trees ;  how  many  are  there  of  both  ? 

18.  Required  the  cost  of  Hunter's  coat,  if  the  cloth 
cost  12  dollars,  and  the  making,  5  dollars. 

19.  Rose  gave  13  cents  to  her  brother,  and  10  to  hei 
sister ;  how  many  cents  did  she  give  away  ? 

20.  Mary's  mother  gave  her  11  apples,  and  her  father 
gave  her  7 ;  how  many  did  they  both  give  her? 

21.  If  I  have  12  pencils,  and  find  5  more,  how  many 
pencils  will  I  then  have  ? 

22.  Russell  gave  3  cents  for  a  top,  4  for  a  whip,  and 
5  for  a  book;  what  did  they  all  cost? 

23.  Louisa  had  5  peaches,  her  mother  gave  her  6,  and 
her  sister  gave  her  7 ;  how  many  had  she  then  ? 

24.  Ruth  bought  5  yards  of  silk  for  a  hat,  7  for  a 
cloak,  and  9  for  a  dress ;  how  many  yards  did  she  buy  ? 

25.  The  head  of  a  fish  is  6  inches  long,  the  tail  8,  and 
the  body  10 ;  what  is  the  length  of  the  fish  ? 

26    Philo  caught  12  fish,  Milo  9,  and  Nero  6;  how 
many  did  they  all  catch  ? 

27.  Howard  shot  11  robins,  Howell  12,  and  Hoyt  13; 
how  many  did  they  all  shoot  ? 

28.  A  man  bought  a  robin  for  10  cents,  a  jay  for  20 


MENTAL  ARITHMETIC.  9 

cents,  -and  a  blue  bird  for  30  cents;  required  the  cost  of 
ail. 

29.  A  sleigh  cost  50  dollars,  a  horse  150  dollars,  and 
a  whip  5  dollars  ;  what  was  the  cost  of  all  ? 

30.  A  lady  gave  6  cents  for   needles,  12  cents  for 
thread,  and  10  cents  for  muslin;  what  was  the  cost  of 
all  ? 

31.  Addie  has  9  roses,  Lizzie  has  15,  and  Amy  had 
20 ;  how  many  have  they  all  ? 

32.  A  man  bought  some  rye  for  25  dollars,  and  some 
wheat  for  26  dollars ;  what  did  he  pay  for  both  ? 

83.  A  merchant  sold  some  rice  for  15  dollars,  some 
sugar  for  17  dollars,  and  some  molasses  for  22  dollars; 
required  the  amount  received. 

34.  How  many  are  4  and  8  ?    5  and  7  ?    3  and  9  ?    5 
and  12  ?    8  and  17  ?    6  and  23?    7  and  15  ?   8  and  26? 

10  and  28  ?  10  and  32  ? 

35.  How  many  are  5  and  21  ?  8  and  23  ?  9  and  31  ? 

11  and  16  ?  3  and  19  ?  8  and  27  ?  10  and  27  ?  11  and 
11?  12  and  12?  12  and  24? 

36.  How  many  are  6  and  16  ?  7  and  17  ?  8  and  18  ? 

9  and  19  ?  10  and  20  ?  11  and  21  ?  12  and  22  ?  13  and 
23?  20  and  30?  21  and  22  ? 

37.  How  many  are  2  and  12  ?  3  and  13  ?  4  and  14  ? 
5  and  15  ?  6  and  26  ?  7  and  27  ?  8  and  28  ?  9  and  29  ? 

10  and  30  ?  11  and  31  ? 

38.  How  many  are  2  and  17  ?  3  and  27  ?  4  and  36  ? 
5  and  46  ?  12  and  41  ?  15  and  29?  24  and  37  ?  43  and 
57?  38  and  65?  56  and  44? 

A  single  thing  is  a  unit.  One  or  more  units  is  a  number. 
Similar  numbers  are  those  in  which  the  units  are  the  same 
Addition  is  the  process  of  finding  the  sum  of  two  or  more 
numbers. 

The  symbol,  -)-,  is  the  sign  of  addition.  It  is  read  plus,  and 
when  placed  between  two  numbers  denotes  that  they  are  to  be 
added  together.  The  symbol,  — ,  is  the  sign  of  subtraction 
It  is  read  minus,  and  when  placed  between  two  numbers  denotes 
that  the  second  is  to  be  subtracted  from  the  first. 


10  MENTAL  ARITHMETIC. 


LESSON  II. 

1.  If  I  have  4  cents,  and  give  2  of  them  away,  how 
many  will  I  have  remaining? 

SOLUTION. — If  I  have  4  cents,  and  give  away  2  cents,  I  will 
have  remaining  the  difference  between  4  cents  and  2  cents,  which 
is  2  cents.  Therefore,  &c. 

2.  Hunter  had  6  apples,  and  gave  2  of  »them  away  ; 
how  many  had  he  remaining  ? 

3.  Morgan  having  9  peaches,  gave  his  sister  3  of  them  : 
how  many  had  he  left  ? 

4.  Ada  culled  10  roses,  and  gave  Lydia  5  of  them; 
how  many  did  she  retain  ? 

5.  A  man  bought  12  lemons,  and  sold  7  of  them;  how 
many  remained  unsold? 

6.  Reuben  finding  13  cents,  spent  8  of  them ;   how 
many  had  he  remaining  ? 

7.  A  watch  was  bought  for  20  dollars,  and  sold  for  27 
dollars ;  what  was  the  gain  ? 

8.  Mary  has  28  pins,  and  Susan  15;  how  many  more 
has  Mary  than  Susan  ? 

9.  A  cow  was  bought  for  27  dollars,  and  sold  for  19 
dollars;  required  the  loss. 

10.  In  a  school  of  40  pupils,  only  29  are  present;  how 
many  are  absent  ? 

11.  A  horse  cost  125  dollars,  and  was  sold  for  140 
dollars ;  required  the  gain. 

12.  William  said  he  found  43  marbles,  and  lost  13  of 
them;  how  many  remained? 

13.  A  farmer  having  27  cows,  sold  18  of  them  ;   how 
many  cows  had  he  remaining  ? 

14.  How  many  are  9  less  5?  11  less  7?  13  less  9?  15 
less  11?  18  less  12? 

15.  How  many  are  16  less  8?  14  less  6?  12  loss  4? 
10  less  2?  14  less  8? 


MENTAL  ARITHMETIC.  11 

16.  Required  the  value  of  12  —  8?  13  —  7?  14--6? 
15  —  5?  16  —  4? 

17.  Inquired  the  value  of  10  —  5?  12—6?  14  —  7? 
1<5  — 8?  18—9? 

18.  Required  the  value  of  11  —  7?  14  —  8?  17—9? 
20  —  10?  23  —  11?  26  —  12? 

19.  Required  the  value  of  6  — 2?  10  —  4?  14—6? 
8  —  8?  22  —  10?  26  —  12? 

20.  What   is   the   value   of  6  +  8—4?   7  +  5  —  6? 
5  +  10  —  12? 

What  is  the  value 

•    21.  Of  7  +  6  — 5?  28.  Of  10  — 6  +  4? 

22.  Of  3 +  8— 9?  29.  Of  16  — 7  +  8? 

23.  Of5  +  6  — 7?  30.  Of  18  — 6  +  3? 

24.  Of  7  +  8  — 9?  31.  Of  15  —  8  +  2? 

25.  Qf9  — 6  +  3?  32.  Of  16— 7  +  8? 

26.  Of8  — 5  +  10?  33.  Of  17  — 9  +  4? 
97.  Of  6— 3  +  12?  34.  Of  22  — 12  +  13? 

"  35.  Silas  having  3  dollars,  found  6;  and  then  lost  4 ; 
how  many  had  he  remaining  ? 

36.  A  boy  having  12  apples,  bought  6  more,  and  then 
sold  8  ;  how  many  had  he  left  ? 

37.  A  man  sold  a  colt  for  25    dollars,    which   is   3 
dollars  more  than  it  cost;  required  the  cost. 

38.  James  had   31  cents,  and   Harry  had  27;    how 
many  had  they  both,  and  how  many  had  James  more  than 
Harry  ? 

39.  John  having  20  peaches,   eat  4,   and  gave   hi? 
sister  6 ;  how  many  had  he  remaining  ? 

40.  Philo  having  12  cents,  spent  7,  and  then  found  9; 
how  many  had  he  then? 

41  A  man  having  a  certain  number  of  cows,  bought 
6,  and  sold  10,  and  then  had  none  left;  how  many  had 
be  at  first  ? 

42.  Morris  having  a  certain  number  of  books,  bough! 
10,  and  giving  30  to  his  sister,  had  none  remaining;  ho\v 
many  books  bad  he  at  first 9 


12  MENTAL  ARITHMETIC. 

43.  Paxton  had  16  peaches,  lie  gave  6  to  James,  and 

7  to  Henry ;  how  many  had  he  remaining  ? 

44.  Edwin  lost  4  cents,  and  found  6,  and  then  had 
1 1 ;  how  many  had  he  at  first  ? 

45.  A  man  sold  13  cows,  then  bought  10,  and  then 
had  1 2 ;  how  many  had  he  at  first  ? 

46.  Two  boys  commenced   playing  with  20   maibles 
Dach,  at  the  close  of  the  game  one  had  16;  how  many 
bad  the  other? 

47.  A  merchant  bought  goods  to  the  amount  of  27 
dollars;  how  must  he  sell  them  to  gain  11  dollars? 

48.  Thomas  had  11  cents,  Susan  gave  him  12,  and 
Walton  gave  him  enough  to  make  his  number  30 ;  how 
many  did  Walton  give  him  ? 

49.  Mr.  A  gave  35  dollars  for  a  case  of  goods,  and 

Iid  4  dollars  for  cartage ;  how  must  he  sell  them  to  gain 
dollars  ? 

50.  How  many  are  4  plus  6  minus  5  ?  7  plus  8  minus 

10  ?  8  plus  12  minus  9  ?  12  plus  13  minus  14  ?  16  plus 
20  minus  30  ?  7  plus  15  minus  12  ? 

51.  How  many  are  7  and  20  minus  13  ?    9  and  13 
oiinus  16  ?  13  and  15  minus  16  ?  18  and  19  minus  20  ? 
16  and  17  minus  18  ?'  20  and  30  minus  40  ? 

52.  How  many  are  6  plus  8  minus  10  ?  7  plus  9  minus 

11  ?  8  plus  10  minus  12  ?  9  plus  11  minus  13  ?  10  plus 

12  minus  14  ?  15  plus  17  minus  19  ? 

53.  How  many  are  4  plus  7  minus  10  ?  5  plus  8  minus 
11?    6  plus  9  minus  12?    7  plus  10  minus  13?    8  plus 
11  minus  14  ?  9  plus  12  minus  15  ? 

54.  How  many  are  6  plus  10  minus  14  ?    7  plus  11 
minus  15  ?    8  plus  12  minus  16?    9  plus  13  minus  17  ? 
10  plus  14  minus  18  ?  11  plus  15  minus  19  ? 

55.  How  many  are  4  plus  24  minus  14?    6- plus  26 
minus  16  ?    7  plus  27  minus  17  ?    5  plus  25  minus  15  ? 

8  plus  28  minus  18  ?  9  plus  29  minus  19  ? 

56.  How  many  are  7  plus  37  minus  27?    10  plus  iiO 
minus  20  ?  11  plus  31  minus  21  ?  12  plus  32  minus  22  ? 

13  plus  33  minus  23  ?  14  plus  34  minus  24  ? 


MENTAL   ARITHMETIC.  18 

57.  How  many  are  4  and  4  minus  2  and  2  ?    6  and  6 
minus  3  and  3  ?    12  and  12  minus  6  and  6  ?    23  and  28 
minus  13  and  13  ?    27  and  27  minus  17  and  17  ? 

58.  How  many  are  2  and  20  taken  from  3  and  30? 
5  and  55  from  6  and  66  ?    4  and  44  from  5  and  55  ?    7 
and  77  from  8  and  88  ?  9  and  99  from  12  and  112  ? 

Subtraction  is  the  taking  of  one  number  from  another.  The 
larger  number  is  called  the  minuend;  the  smaller,  ^the  subtra- 
hend; the  result,  the  difference  or  remainder. 


LESSON  III. 

1.  What  cost  4  apples,  at  2  cents  apiece? 

SOLUTION. — If  one  apple  cost  2  cents,  4  apples  will  cost  4 
times  2  cents,  which  are  8  cents.     Therefore,  £c. 

2.  What  cost  5  oranges,  at  3  cents  apiece? 

3.  At  5  cents  each,  what  will  4  melons  cost? 

4.  What  will  8  sheep  cost,  at  6  dollars  apiece  ? 

5.  At  3  dimes  each,  what  will  12  turkeys  cost? 

6.  What  cost  7  cows,  at  20  dollars  each  ? 

7.  What  cost  12  horses,  at  200  dollars  apiece  ? 

8.  At  30  cents  apiece,  what  cost  11  knives  ? 

9.  What  cost  12  books,  at  20  cents  apiece  ? 

10.  At  4  dimes  each,  what  cost  9  ducks  ? 

11.  How  far  will  a  man  travel  in  9  days,  at  the  rate 
of  12  miles  a  day? 

12.  A  boy  has  13  apples,  and  5  times  as  many  peaches ; 
required  the  number  of  peaches, 

13.  Mary  has  14  apples,  and  John  has  7  times  as  many 
how  many  has  John  ? 

14.  A  has  13  dollars,  and  B  has  9  times  as  many; 
what  number  have  they  both  ? 

15.  A  farmer  sold  4  horses,  and  then  bought  6  times 
as  many ;  how  many  did  he  buy  ? 

2* 


14  MENTAL  ARITHMETIC. 

16.  How  many  dimes  must  be  paid  for  9  books,  at  the 
rate  of  14  dimes  each  ? 

17.  Amos  saw  13  flocks  of  pigeons,  with  20  pigeons  in 
each  flock ;  how  many  pigeons  did  he  see  ? 

18.  A  merchant  having  10  melons,  sold  6,  and  then 
bought  5  times  as  many  as  he  sold;  how  many  had  he 
then  ? 

19.  B  having  20  sheep,  sold  12,  and  then  bought  4 
times  as  many  as   remained;    how   many  did   he    then 
have  ? 

20.  A  boy  borrowed  6  cents  from  a  friend,  and  then 
earned  7  times  as  much  as  he  borrowed;  how  many  cents 
had  he  then? 

21.  Mary  found  10  pins,  and  then  bought  8  times  as 
many  as  she  found ;  how  many  had  she  then  ? 

22.  What  will  7  yards  of  muslin  cost,  at  8  cents  a 
yard? 

23.  How  much  cost  11  yards  of  ribbon,  at  the  rate  of 
11  cents  a  yard  ? 

24.  If  3  men  can  do  a  piece  ot  work  in  5  days,  how 
long  wi!1  it  take  1  man  to  do  it  ? 

,  25.  If  5  men  can  do  a  piece  of  work  in  20  days,  how 
long  will  it  require  1  man  to  do  it? 

26.  How  far  will  a  man  travel  in  9  hours,  at  the  rate 
of  3  miles  an  hour  ? 

27.  In  an  orchard  there  are  11  rows  of  trees,  and  20 
trees  in   each  row;    how  many  trees  are  there  in  the 
orchard  ? 

28.  If  9  men  mow  a  field  of  grass  in  14  days,  how 
long  will  it  take  1  man  to  mow  it  ? 

29.  Mason  earned  10  dollars  a  week,  and  paid  3  dol- 
lars a  week  for  his  board;  how  much  would  he  save  in  5 
weeks  ? 

30    If  1  melon  is  worth  3  peaches,  how  many  peaches 
are  7  melons  worth  ? 

31.  Passmore  earned  3  dollars  a  day,  and  paid  1  dol« 
lar  for  his  board ;  how  much  could  he  save  in  a  week  ? 

32.  Morton  earned  30  dollars  a  month,  he  paid  o  del- 


MENTAL    ARITHMETIC.  15 

lars  a  week  for  board,  and  1  dollar  for  other  expenses : 
how  much  could  he  save  in  a  year  ? 

33.  Thomas  travels  5  miles  an  hour,  and  John,  3; 
how  much  farther  will  Thomas  travel  in  12  hours  than 
John? 

84.  How  many  are  6  times  6,  plus  6?  5  times  8,  plus 
8  ?  7  times  10,  plus  10  ?  4  times  11,  plus  11  ?  8  times 

I,  plus  4?  5  times  9,  plus  9  ? 

35.  How  many  are  3  times  4,  plus  5  ?  6  times  7,  plus 

8  ?    8  times  9,  plus  10  ?   9  times  10,  plus  11  ?    10  times 

II,  plus  12  ?  11  times  12,  plus  13  ? 

36.  How  many  are  4  times  4,  minus  5?    5  times  7, 
minus  6?     7  times  9,  minus  8?     8  times  10,  minus  9? 

9  times  12,  minus  11  ?  11  times  12,  minus  13  ? 

37.  How  many  are  7  and  7  times  8  ?  5  and  5  times  6  ? 
8  and  8  times  4  ?    6  and  6  times  7  ?    9  and  9  times  11  ? 

10  and  10  times  12? 

38.  How  many  are  4  and  5  and  4  times  5  ?    5  and  6 
and  5  times  6  ?    6  and  7  and  6  times  7  ?    7  and  8  and  7 
times  8  ?  8  and  9  and  8  times  9  ? 

39.  How  many  are  23  minus  2  times  3  ?    34  minus  3 
times  4  ?    25  minus  2  times  5  ?    37  minus  3  times  7  ? 
49  minus  4  times  9  ?  , 

40.  B  and  C  start  from  the  same  place,  and  travel  in 
opposite  directions,  B  at  the  rate  of  5,  and  C  4  miles  an 
hour ;  how  far  are  they  apart  in  6  hours  ? 

41.  Two  men  start  at  the  same  place,  and  travel  in 
the  same  direction,  the  one  7,  and  the  other  5  miles  an 
hour ;  how  far  will  they  be  apart  in  10  hours  ? 

Multiplication  is  the  process  of  taking  one  number  as  many 
times  as  there  are  units  in  another.  The  number  repeated  u 
the  multiplicand;  the  number  of  times  it  is  repeated  is  the 
multiplier;  the  result  is  the  product. 

The  symbol,  X'  ^s  tne  s^gn  °f  multiplication;  when  placed 
between  two  numbers  it  denotes  that  they  are  to  be  multiplied 
together.  The  symbol,  -5-,  is  the  sign  of  division  :  when 
placed  between  two  numbers  it  denotes  that  the  one  before  ii 
is  to  be  divided  by  the  one  following  it. 


16  MENTAL    ilUTHMETIC. 


LESSON  IV. 

1.  How  many  2's  are  there  in  6  ? 

SOLUTION. — In  6  there  are  3  two's,  because  3  times  2  are  6. 

2.  How  many  2's  in  8  ?  in  10  ?  in  12  ?  in  16  ? 

3.  How  many  3's  in  6  ?  in  9  ?  in  15  ?  in  18  ? 

4.  How  many  4's  in  8  ?  in  16  ?  in  24  ?  in  28  ? 

5.  How  many  5's  in  15  ?  in  20  ?  in  30  ?  in  45  ? 
6  How  many  6's  in  18  ?  in  30  ?  in  40  ?  in  48  ? 

7.  How  many  7's  in  21?  in  35  ?  in  28  ?  in  42  ? 

8.  How  many  8's  in  32?  in  40  ?  in  56  ?  in  72  ? 

9.  How  many  9's  in  36  ?  in  45  ?  in  63  ?  in  81  ? 

10.  How  many  10's  in  40  ?  in  50  ?  in  70  ?  in  90  ? 

11.  How  many  ll's  in  44  ?  in  33  ?  in  66  ?  i    88  ? 

12.  How  many  12's  in  48  ?  in  72  ?  in  96  i  in  132  '< 

13.  24  contains  3  how  many  times?  4  ?  6?  8?  12? 

SOLUTION. — 24  contains  3,  eight  times,  smo-e  8  times  3  are  24 

14.  30  contains  2  now  many  times?  3?  5?  6?  10? 

15.  36  contains  3  how  many  times?  4?  6?  9?  12? 

16.  40  contains  4  how  many  times  ?  5  ?  8  ?  10  ?  20  ? 

17.  60  contains  5  how  many  times?  6?  10?  12?  20? 
IS.  80  contains  4  how  many  times?  8?  10?  20?  40? 

19.  90  contains  3  how  many  times?  6?  9?  10?  301 

20.  100  contains  4  how  many  times?  5?  10?  20?  25? 

21.  How  many  times  3  are  6  times  2  ? 
22    How  many  times  5  are  4  times  10  ? 
23.   II o^  many  times  7  are  3  times  14  ? 

24  How  many  times  8  are  5  times  6,  -f  2  t 

2 a  4  times  7,  -j-  2,  are  how  many  times  6  ? 

20  6  times  8,  —  8,  are  how  many  times  10  ? 

27.  7  times  9,  —  3,  are  how  many  times  12  ? 

28.  8  times  7,  -(-  4,  are  how  many  times  4  ? 

29.  10  times  9,  —  6,  are  how  many  times  7  ? 
BO  9  times  8,  +  8,  are  how  many  times  8? 


MENTAL  ARITHMETIC.  17 

81.  5  times  12,  —  4,  are  how  many  times  1  .  ? 
32    4  times  11,  -f-  6,  are  how  many  times  5  ? 

33.  5  limes  6,  -f-  10,  —  8,  are  how  many  times  8  ? 

34.  7  times  8,  —  6,  -f-  5,  are  how  many  times  5  ? 

35.  8  times  9,  -f-  8,  —  10,  are  how  many  times  10  ? 

36.  9  times  9,  -j~  9,  —  2,  are  how  many  times  8  ? 

37.  8,  and  4,  and  2,  and  3,  and  5,  and  6,  less  12,  an 
h  >w  many? 

38.  Five,  and  6,  and  4,  and  7,  and  8,  and  10,  less  20, 
are  how  many  ? 

39.  Four,  and  7,  and  6,  and  5,  and  3,  and  2,  less  17 , 
are  how  many  ? 

40.  Twelve,  multiplied  by  2,  divided  by  3,  multiplied 
by  4,  divided  by  8,  plus  6,  are  how  many? 

41.  Three   times  8,   increased   by  6,   divided  by  3, 
diminished  by  5,  are  how  many  ? 

42.  How  many  times  5  times  6,  increased  by  10,  is  6 
times  10;  increased  by  20  ? 


LESSON  V, 

1.  At  3  cents  each,  how  many  melons  can  I  buy  fof 
12  cents? 

SOLUTION. — If  1  melon  cost  3  cents,  for  12  cents  I  can  buy  as 
many  melons  as  3  is  contained  times  in  12,  which  are  4.  There- 
fore, &c 

2  At  2  dimes  apiece,  how  many  books  can  I  buy  foi 
3  d.mes? 

3.  At  4  cents  a  yard,  how  many  yards  of  ribbon  can  I 
buy  for  16  cents  ? 

4.  How  many  apples  can  I  buy  for  21  cents,  at  3  cents 
apiece  ? 

5.  How  many  yards  of  ribbon,  at  6  cents  a  yard,  can 
be  bought  for  42  cents? 


IS  MENTAL  ARITHMETIC. 

6.  A  man  gave  50  dollars  for  sheep,  at  the  rate  o£  * 
doLars  a  head ;  how  many  did  he  buy  ? 

7.  If  a  man   travel  4  miles  an  hour,  how  long  will  it 
take  him  to  travel  48  miles  ? 

8.  A  man  gave  7  boys  56  cents;  how  much  did  euob 
boy  receive? 

9  If  8  apples  cost  24  cents,  how  much  will  one  apple 
cost? 

10.  A  farmer  received  88  dollars  for  sheep,  at  the  rate 
of  8  dollars  each  ;  how  many  did  he  sell  ? 

11.  How  many  kegs,  of  9  gallons  each,  can  be  rilled 
from  a  hogshead,  containing  63  gallons  of  molasses? 

12.  How  many  days  must  a  man  worl^  to  earn  44  dol- 
lars,  at  the  rate  of  4  dollars  a  day  ? 

13.  How    many  melons,  at   7   cents    apiece,  may  be 
bought  for  84  cents  ? 

14.  How  many  oranges,  at  3  cents  each,  may  be  had 
for  5  lemons,  worth  6  cents  each  ? 

15.  How  many  yards  of  lace,  at  6  cents  a  yard,  may 
be  bought  for  3  yards  of  muslin,  at  12  cents  per  yard  ? 

16.  If  a  man  digs  10  yards  of  ditch,  for  8  dimes  a 
yard,  how  many  bushels  of  rye,  at  4  dimes  a  bushel,  will 
pay  him  ? 

17.  How  many  pounds  of  meat,  at  6  cents  a  pound, 
will  cost  as  much  as  9  yards  of  ribbon,  at  8  cents  a  yard  ? 

18.  How  many  boxes  of  wafers,  worth  6  cents  a  box, 
may  be  bought  for  12  sheets  of  paper,  worth  2  cents  a 
sheet? 

19.  How  much  wheat,  at  6  dimes  a  bushel,  may  be 
purchased  for  12  bushels  of  corn,  worth  5  dimes  a  bushel  ? 

20.  How  many  knives,  at  3  dimes  each,  can  I  buy  foi 
I  dimes  in  money,  and  7  boxes  of  raisins,  at  4  diunis 
such  ? 

21.  If  flour  is  worth  8  dollars  per  barrel,  how  man} 
barrels  can  be  bought  for  3  dollars,  and  7  barrels  of  fish, 
at  11  dollars  per  barrel  ? 

22  A  man  gave  9  pencils,  worth  5  cents  each,  for  3 
packages  of  envelopes  worth  11  cents  each;  what  did  he 
lose? 


MENTAL    ARITHMETIC.  19 

23.  How  many  lamps  at  7  dimes  each,  can  be  bought 
for  9  dimes  in  money,  and  6  dozen  eggs,  at  2  dimes  per 
dozen  ? 

24.  How  many  are  12  plus  6  divided  by  6  ?  25  plus  5 
divided  by  5  ?  32  plus  8  divided  by  8  ?  42  plus  7  divided 
by  7?  45  plus  9  divided  by  9? 

25.  How  many  are  16  and  4  divided  by  4  ?  21  and  7 
divided  by  Y?  44  and  11  divided  by  11?    60  and  12 
divided  by  12?  96  and  8  divided  by  8? 

26.  How  many  are  90  minus  9  divided  by  9  ?     SO 
minus  8  divided  by  8  ?  100  minus  4  divided  by  4  ?  110 
minus  10  divided  by  10  ?  144  minus  12  divided  by  12  ? 

27.  How  many  are  6  times  8  divided  by  4  ?    7  times 
9  divided  by  3  ?    10  times  6  divided  by  12  ?    9  times  9 
divided  by  3  :    12  times  10  divided  by  5  ? 

28.  How  many  are  2  times  22  divided  by  11  ?  4  times 
14  divided  by     ?    3  times  15  divided  by  5  ?    6  times  16 
divided  by  8  ?  8  times  18  divided  by  12  ? 

29.  How  many  are  48  divided  by  6,  plus  8  ?  30  divided 
by  5,  plus  6  ?  Si  8  divided  by  7,  plus  17  ?   50  divided  by 
5,  plus  15  ?  35  divided  by  7,  plus  17  ?  84  divided  by  12, 
plus  22  ? 

30.  Twice  a  number,  -f-  3  times  the  number,  —  4  times 
the  number,  -)-  2  times  the  number,  equals  how  many 
times  the  number? 

NOTE. — With  young  pupils  this  and  the  following  problems 
should  be  read  slowly,  that  the  pupils  may  solve  them  as  the 
teacher  reads.  More  advanced  pupils  should  be  required  to 
enunciate  them  after  the  teacher  has  read  them,  and  then  solve 
them. 

31.  Three  times  a  number,  —  2  times  the  number,  -|- 
4  times  the  number,  —  5  times  the  number,  equals  how 
many  times  the  number  ? 

32.  Eour  times  Susan's  age,  -f-  3  times  her  age,  —  5 
times  her  age,  -f  2  times  her  age,  —  3  times  her  age, 
equals  13  years  ;  how  old  is  Susan  ? 

3o.  Three  times  Henry's  number  of  marbles,  -f-  twice 
his  number,  —  3  times  his  number,  -f-  4  times  his  num- 
ber,--5  times  his  number,  equals  15;  how  many  has  he  ? 


20  MENTAL  ARITHMETIC. 

34  Eight  times  a  number,  -j-  by  4,X  by  6,  -5- by  8,  X 
by  5,  —  by  10,  equals  how  many  times  the  number  ? 

35.  Five  times  a  number,  X  by  4,  -4-  by  10,  X  by  6, 
-f-by  4,  X  by  2,  equals  how  many  times  the  number? 

36.  Think  of  any  number,  multiply  it  by  6,  divide  by 
8,   multiply  by  2,   divide  by  4,  add    10,  subtract   tli€ 
number  thought  of,  divide  by  5,  and   the  quotient   ie 
what  ? 

'•>! .  Think  of  a  number,  multiply  it  by  10,  divide  it 
by  5,  multiply  by  3,  divide  by  6,  add  10,  subtract  the 
number  thought  of,  divide  by  8,  and  the  quotient  i& 
what  ? 

38.  Three  times  Mary's  age,  multiplied  by  6,  divided 
by  9,  multiplied  by  4,  divided  by  8,  equals  16  years; 
how  old  is  Mary  ? 

39.  Think  of  a  number,  multiply  it   by  5,  multiply 
that  by  4,  divide  the  product  by  10,  multiply  by  6.  di- 
vide by  3,  add  30,  subtract  4  times  the  number,  divide 
by  5^  and  name  the  quotient. 

40.  What  is  Division  ?     Name  and  define  each  term 
employed.     What  kind  of  numbers  may  each  term  bo 
respectively  ? 


SECTION    II. 

LESSON  I, 

1 .  If  I  divide  an  apple  into  two  equal  parts,  what  is 
'i>ae  of  these  parts  called?  What  are  2  of  these  parts 
called  ? 

2  How  many  halves  of  an  apple  in  one  apple? 

3  What  is  1  half  of  6 

SOLUTION — 1  half  of  6  is  3,  because  2  times  3  are  6 


.  MENTAL  ARITHMETIC.  21 

4  Wnatislhalfof4?  of  8?  of  10?  of  12  t 

5  What  is  1  half  of  14  ?  of  16  ?  of  18  ?  of  20  ? 

6.  What  is  1  half  of  22  ?  of  26  ?  of  28  ?  of  32  ? 

7.  If  1  yard  of  cloth  cost  16  cents,  what  will  1  half  of 
a  yard  cost  ? 

8    If  1  pound  of  sugar  cost  10  cents,  what  will  1  half 
of  a  pound  cost  ? 

9.  Paul  having  20  apples,  gave  1  half  of  them  to  bis 
brother ;  how  many  did  he  give  away  ? 

10.  Thompson  bought  24  cows,  and  sold  1  half  of 
them ;  how  many  did  he  retain  ? 

11.  Phebe  had  80  peaches,  and  gave  1  half  of  them 
away ;  how  many  remained  ? 

12.  If  I  divide  an  apple  into  3  equal  parts,  what  is 
one  of  these  parts  called  ? 

13.  What  are  2,  and  3  of  these  parts  called? 

14.  How  many  thirds  in  one  apple  ? 

15.  What  is  1  third  of  6?  of  9  ?  of  12?  of  15? 

16.  What  is  1  third  of  21?  of  24?  of  30?  of  36?  ' 

17.  James  had  30  cents,  and  lost  1  third  of  them; 
how  many  did  he  lose  ? 

18.  Henry  had  39  pears,  and  Thomas  had  1  third  as 
many  •  how  many  had  Thomas  ? 

19.  Lucy  had  21  pins,  and  gave  Mary  1  third  of  them; 
how  many  did  Mary  receive? 

20.  Matthew  bought  24  oranges,  and  gave  1  third  of 
them  to  Stephen ;  how  many  remained  ? 

21.  A  bought  42  cows,  and  sold  1  third  of  them  to 
B;   how  many  had  he  remaining  ? 

22.  What  are  2  thirds  of  9  ? 

SOLUTION.— 1  third  of  9  is  3,  and  if  1  third  of  9  is  3,  2  thirds 
of  9  are  2  times  3,  which  are  6.     Therefore,  &c. 

23.  What  are  2  thirds  of  6?  of  12?  of  15?  of  IS? 

24.  What  are  2  thirds  of  24?  of  30?  of  27  ?  of  33? 

25.  John  had  21  cents,  and  gave  2  thirds  of  them  to 
Sarah ;  how  many  cents  did  Sarah  receive  ? 

3 


22  MENTAL  ARITHMETIC.  « 

26  Having  27  peaches,  I  sold  2  thirds  of  them ;   ho\* 
many  did  I  sell  ? 

27  Henry  gave  his  sister  2  thirds  of  39  oranges;  how 
many  did  he  retain  ? 

28.  Hiram   lost  2  thirds  of  36   dollars;    how   much 
money  had  he  remaining  ? 

29.  Chandler,  having  30  pears,  gave  1  third  to  Maiia; 
and  1  third  to  Jane;  how  many  remained? 

30.  If  I  divide  an  apple  into  4  equal  parts,  what  are 
1.  2,  and  3  of  these  parts  ? 

31.  How  many  fourths  of  an  apple  in  a  whole  apple  ? 
•32.  What  is  1  fourth  of  4?  of  8?  of  20?  of  32? 

33.  What  is  1  fourth  of  12?  of  24?  of  16?  of  48? 

34.  What  are  2  fourths  of  24?  of  16?  of  28?  of  36? 

35.  What  are  2  fourths  of  12?  of  20?  of  40?  of  48? 

36.  What  are  3  fourths  of  20?  of  24?  of  12?  of  16? 

37.  What  are  3  fourths  of  16?  of  28?  of  40?  of  52? 

38.  Jacob  having  44  pens,  sold  2  fourths  of  them  to 
Joseph ;  how  many  pens  did  Joseph  receive  ? 

39.  If  a  yard  of  cloth  cost  8  dollars,  what  will  3  fourths 
of  a  yard  cost  ? 

40.  A  boy  sold  2  fourths  of  28  pears;  how  many  had 
he  remaining? 

41.  Samson  gave  his  brother  1  fourth,  and  his  sistei  2 
fourths  of  28  oranges;  how  many  did  each  receive? 

42.  Marion  found  2  fourths  of  28  cents,  and  then  lost 
3  fourths  of  16  cents ;  how  many  remained  ? 

43.  A  having  24  plums,  gave  1  half  of  them  to  B, 
and  1  third  to  C ;  how  many  had  he  left  ? 

44.  Harleigh  is  24  years  of  age,  a'nd  Townsend  is  3 
fourths  as  old;  how  old  is  Townsend  ? 

45.  A  farmer  had  36  sheep,  of  which  A  bought  1 
half,  and  a  dog  kil-led  1  third ;  how  many  remained  ? 

46.  A  merchant   having   40  barrels  of  flour,  sold  3 
fourths  of  them,  and  then  bought  1  third  as  many  as  he 
sold ;  how  many  had  he  then  ? 


MENTAL  ARITHMETIC.  23 


LESSON  II, 

1.  If  you  divide  an  orange  into  5  equal  parts,  what 
are  1    2,  3,  and  4  of  these  parts  called  ? 

2.  How  many  fifths  in  one  orange? 

8.  What  is  1  fifth  of  10  ?  of  25  ?  of  15  ?  ot  30  ? 

4.  What  are  2  fifths  of  15  ?  of  30  ?  of  45  ?  of  60  ? 

5.  What  are  2  fifths  of  65  ?  of  35  ?  of  40  ?  of  50  ? 

6.  What  are  3  fifths  of  10  ?  of  30  ?  of  25  ?  of  55  ? 

7.  What  are  4  fifths  of  20  ?  of  50  ?  of  60  ?  of  100  ? 

8.  Mary  has  15  oranges,  and  Rachael  2  fifths  as  many; 
how  many  has  Rachael  ? 

9.  Susan's  age  is  25  years,  and  her  sister  is  4  fifths  as 
old ;  how  old  is  her  sister  ? 

10.  Rowland  is  35  years  of  age,  and  his  wife  is  3  fifths 
as  old }  how  old  is  his  wife  ? 

11.  A  horse  cost  100  dollars,  and  a  sleigh  3  fifths  as 
much ;  required  the  cost  of  the  sleigh. 

12.  A  man  having  40  sheep,  lost  20,  and  found  only 
3  fifths  of  them  ]  how  many  remained  ? 

13.  A  man  having  50  cows,  sold  4  fifths  of  them,  and 
then  bought  4  fifths  as  many  as  he  sold";  how  many  had 
he  then  ? 

14.  4  fifths  of  200  dollars,  is  2  times  what  Emily's 
watch  cost;  what  was  the  cost  of  the  'watch? 

15.  If  you  divide  a  melon  into  6  equal  parts,  what  are 
1,  2,  3,  4,  and  5  of  these  parts  called? 

16    How  many  sixths  are  there  in  a  single  thing? 

17.  What  are  2  sixths  of  24?  of  18?  of  36?  of  60? 

18.  What  are  3  sixths  of  12  ?  of  42?  of  30?  of  66? 

19.  What  are  4  sixths  of  6  ?  of  36  ?  of  48  ?  cf  54  ? 

20.  What  are  5  sixths  of  18?  of  54?  of  24?  of  72? 

21.  What  will  2  sixths  of  12  yards  of  tape  cost,  at  3 
cents  a  yard? 

22.  llaub  having  48  pens,  sold  3  sixths  to  Frescoln. 
and  2  sixths  tc  Morgan ;  how  many  did  he  sell  tc  both  ? 


24  MENTAL  ARITHMETIC. 

23.  Whai  will  5  sixths  of  36  yards  of  cloth  cost,  at 
the  rate  of  2  dollars  a  yard  ? 

24.  Warren  had  12  marbles,  and  Oliver  had  5  sixths 
as  many,  lacking  4;  how  many  had  Oliver? 

25.  How  many  yards  of  cloth  can  you  buy  for' 20  dol- 
lars, if  1  yard  cost  2  fifths  of  10  dollars? 

26.  Dana  having  60  peaches,  gave  2  sixths  of  them  to 
Barton,  and  3  sixths  to  Benton;  how  many  remained? 

27.  If  1  yard  of  linen  cost  5  sixths  of  36  cents,  how 
many  yards  can  you  buy  for  3  fourths  of  80  cents? 

28.  2  thirds  of  30  dollars,  increased  by  2  thirds  of  60 
dollars,  is  10  dollars  less  than  A's  money;  required  A's 
money. 

29.  Frazier  having  40.  pens,  gave  Brown  10,  and  Seal 
2  sixths  of  the  remainder ;  how  many  had  he  left  ? 

30.  If  a  melon  be  divided  into  7  equal  parts,  what  are 
1,  2,  3,  4,  5,  and  6  of  these  parts  called? 

31.  How  many  sevenths  are  there  in  one  ? 

32.  What  is  1  seventh  of  21  ?  of  28  ?  of  42  ?  of  56  ? 

33.  What  are  2  sevenths  of  28  ?  of  49  ?  of  63  ?  of  70  ? 

34.  What  are  3  sevenths  of  14  ?  of  35  ?  of  49  ?  of  28  ? 

35.  What  are  4  sevenths  of  70  ?  of  77  ?  of  63  ?  of  84  ? 

36.  What  are  5  sevenths  of  77  ?  of  91  ?  of  42  ?  of  140  ? 

37.  What  are  6  sevenths  of  35  ?  of  42  ?  of  49  ?  of  28  ? 

38.  4  sevenths  of  21,  are  how  many  times  3,  4,  and  6? 

39.  6  sevenths  of  63,  are  how  many  times  3,  6,  and  9? 

40.  A  watch  was  bought  for  70  dollars,  and  sold  for  6 
sevenths  of  its  cost ;  required  the  loss  ? 

41.  If  1  half  of  4  yards  of  cloth  cost  10  dollars,  what 
will  1  fifth  of  10  yards  cost? 

42.  A  gave  70  dollars  for  a  watch,  3  sevenths  as  much 
for  a  chain,  and  sold  them  both  for  90  dollars ;  required 
tne  loss. 

43.  3  sevenths  of  56  dollars  is  6  dollars  more  than  1 
third  of  a4oad  of  hay  cost;  what  will  3  loads  cost,  at  the 
same  rate  ? 

44.  4  sevenths  of  42  dollars  is  12  dollars  less  than  1 
half  of  a  building  lot  cost;  required  the  cost  of  the  lot 


MENTAL  ARITHMETIC.  25 

45.  Richard  had  360  dollars,  1  third  of  which  he  spent 
for  a  horse,  1  fourth  for  a  watch,  and  1  sixth  for  a  sleigh ; 
how  much  had  he  remaining  ? 

46.  Mr.  A  having  140  dollars,  gave  3  sevenths  of  it  to 
the  poor,  and  lost  3  fourths  of  the  remainder;  how  much 
then  remained  ? 


LESSON  III. 

1.  If  anything  be  divided  into  8  equal  parts,  what  is 
one  of  these  parts  called  ? 

2.  What  are  2,  3,  4,  5,  6,  and  7  of  these  parts  called, 
and  how  many  eighths  in  a  unit  ? 

3.  What  is  1  eighth  of  24?  48?  72?  88? 

4.  What  are  2  eighths  of  32?  40?  56?  72? 

5.  What  are  3  eighths  of  16?  64?  80?  96? 

6.  What  are  5  eighths  of  8  ?  24?  48?  64? 

7.  What  is  1  half  of  3  ?  5?  7?  9?  11? 

8.  What  is  1  third  of  5?  7?  8?  10?  13? 

9.  What  are  2  thirds  of  8?  14?  16?  17?  19? 

10.  What  are  3  fifths  of  9  ?  21?  17?  24?  31? 

11.  What  are  5  eighths  of  7?  10?  14?  18?  20? 

12.  2  eighths  of  24  are  how  many  times  3  ? 

13.  3  eighths  of  40  are  how  many  times  5  ? 

14.  4  eighths  of  80  are  how  many  times  8  ? 

15.  5  eighths  of  56  are  how  many  times  7  ? 

16.  6  eighths  of  64  are  how  many  dmes  12  ? 

17  7  eighths  of  72  are  how  many  times  3  ? 

18  3  eighths  of  32  are  how  many  times  1  third  of  12  ? 

19.  6  eighths  of  40  are  how  many  times  1  fourth  of  24  ? 

20.  4  eighths  of  4    are  how  many  times  2  thirds  of  18  ? 

21  7  eighths  of  96  are  how  many  times  4  fifths  of  10  ? 

22  3  eighths  of  56  are  how  many  times  5  sixths  of  42  ? 

23.  5  sevenths  of  28  are  how  many  times  5  sevenths 
of  14? 

24.  2  thirds  of  27  are  how  many  times  3  fourths  of  12  j 
3* 


26  MENTAL  ARITHMETIC 

25.  If  a  single  thing  be  divided  into  9  equal  parts, 
what  are  1,  2,  8,  4,  &c.,  of  these  parts  called? 

26.  What  are  2  ninths  of  18?  27?  45?  36? 

27.  What  are  3  ninths  of  63?  72?  81?  27? 

28.  What  are  4  ninths  of  9  ?  36?  54?  81? 

29.  What  are  5  ninths  of  54  ?  72?  63?  27? 

30.  What  are  6  ninths  of  81  ?  18?  36?  90? 
31    What  are  7  ninths  of  18  ?  99?  27?  108? 

32.  3  times  6,  and  2  thirds  of  6,  are  how  many? 

33.  4  times  12,  and  3  fourths  of  12,  are  how  many? 

34.  5  times  10,  and  3  fifths  of  10,  are  how  many? 

35.  6  times  12,  and  3  sixths  of  12,  are  how  many? 

36.  5  times  7,  and  4  sevenths  of  7,  are  how  many? 

37.  9  times  8,  and  5  eighths  of  8,  are  how  many  ? 

38.  2  times  18,  and  7  ninths  of  18*  are  how  many? 

39.  2  ninths  of  18  are  how  many  times  2  thirds  of  3  ? 

40.  5  ninths  of  27  are  how  many  times  5  sixths  of  6  ? 

41.  6  ninths  of  54  are  how  many  times  4  fifths  of  15  ? 

42.  3  ninths  of  72  are  how  many  times  2  eighths  of  16  ? 

43.  7  eighths  of  24  are  how  many  times  7  eighths 
of  8? 

44.  5  sixths  of  48  are  how  many  times  5  sevenths 
of  14? 

45.  4  sevenths  of  56  are  how  many  times  8  ninths 
of  18? 

46.  A  bought  15  horses,  and  sold  6  of  them,  and  then 
lacked  4  of  having  20 ;  how  many  had  he  at  first  ? 

47.  Hiram  and  Oliver  had  each  26  cents ;  after  Hiram 
had  given  Oliver  10,  and  Oliver  had  given  Hiram  6,  how 
many  had  each  ? 

48.  A  farmer  having  48  bushels  of  oats,  sold  4  sixths 
of  them  to  one  man,  and  1  fourth  to  another ;  how  many 
bushels  did  he  sell  to  each  ? 

49.  A  bought  60  cows,  and  sold  2  sixths  of  them  to 
B,  and  3  times  2  tenths  of  the  remainder  to  C  j  how 
many  then  remained  Jf 


MENTAL  ARITHMETIC  27 


LESSON  IV, 

1     Harry  gave  1  third  of  an  apple  to  his  trother,  and 
2  thiids  to  his  sister;  how  much  did  he  give  away? 

2.  Matthew  gave  2  fifths  of  a  peach  to  Elias,  and  8 
fifths  to  Morris;  how  much  did  he  give  to  both  ? 

3.  If  I  give  3  sevenths  of  a  melon  to  Harry,  and  1 
sevenths  to  Harvey,  how  much  do  I  give  away? 

4.  Fanny  eat  3  eighths  of  a  quart  of  chestnuts  yester- 
day, and  4  eighths  to-day;  how  many  did  she  eat  in  all? 

5.  Ella  gave  1  fourth  of  a  melon  to  Phebe,  2  fourths 
to  Carrie,  and  3  fourths  to  Kate ;  how  much  did  she  give 
away? 

6.  Philip  gave  2  sixths  of  a  dollar  to  Jane,  3  sixths  to 
Sarah,  and  5  sixths  to  Eliza;  how  much  did  he  give 
away? 

7.  Willis  lost  7  eighths  of  a  dollar,  and  had  9  eighths 
emaining  j  how  much  had  he  at  first  ? 

8.  Matthew  lost  6  eighths  of  a  dollar  from  one  pocket, 
and  7  eighths  from  the  other,  and  had  5  eighths  remain- 
ing ;  how  much  had  he  at  first  ? 

9.  Dora  gave  3  ninths  of  a  pound  of  raisins  to  Ella, 
and  7  ninths  to  Daisy,  and  then  had  3  ninths  remaining; 
how  many  had  she  at  first  ? 

10    What  is  the  sum  of  1  fourth,  3  fourths,  5  fourths, 
7  fourths,  and  9  fourths  ? 

11.  Jane  had  7  eighths  of  a  pound  of  candies,  and  gave 
Maria  5  eighths ;  how  many  eighths  remained  ? 

12.  Frank  had  6  sevenths  of  a  melon,  and  gave  Abratfl 
4  sevenths ;  how  much  remained  ? 

13.  Louisa  having  10  eighths  of  a  dollar,  gave  Lizzie 
7  eighths  of  it ;  how  much  remained  ? 

14.  What  is  the  difference  between  5  sevenths,  and  the 
sum  of  4  sevenths  and  6  sevenths  ? 

15     Sallie  having  24  pears,  gave   Buela  2    eighths, 


28  MENTAL  ARITHMETIC. 

Amanda  3  eighths,  and  James  1  eighth  of  them  ;   hew 
many  remained  ? 

16.  Rufus  having  1  third  of  a  quart  of  chestnuts, 
bought  4  thirds  of  a  quart,  and  then  sold  1  quart ;  what 
part  of  a  quart  remained  ? 

17.  Peter  having  5  sixths  of  a  bushel  of  apples,  sold 
sixths,  and  then  bought  2  sixths  of  a  bushel;   how 

many  sixths  had  he  then  ? 

18.  What   is  the  difference  between  the  sum   of  3 
eighths  and  7  eighths,  and  the  sum  of  4  eighths  and  5 
eighths  ? 

19.  A  bought  20  sheep,  and  sold  2  tenths  of  them  to 
B,  3  tenths  to  C,  and  4  tenths  to  D ;  how  many  sheep 
remained  ? 

20.  A  lady  having  36  yards  of  tape,  sold  6  ninths  of 
it  to  one  person,  and  3  ninths  to  another;   how  much 
had  she  then  ? 

21.  Mariana  had  3  fourths  of  a  pint  of  nuts,  Elva  had 
twice  as  many,  and  Ezra  3  times  as  many;  how  many 
had  they  all  ? 

22.  A  bought  4  ninths  of  a  bushel  of  wheat,  and  K 
bought  3  times  as  much ;  how  much  did  B  buy  ? 

23.  If  1  yard  of  cloth  cost  5  sixths  of  a  dollar,  what 
will  6  yards  cost  ? 

24.  At  7  ninths  of  a  dollar  each,  what  will  9  turkeys 
cost  ? 

25.  What  is  the  sum  of  5  times  3  ninths,  and  3  times 
5  ninths  ? 

26.  Mary  having  11  fifths  of  a  melon,  gave  2  fifths 
to  Sarah,  and  twice  as  much  to  Sophia;  how  much  re 
luaiued? 

27.  How  much  greater  is  7  times  2  sevenths,  than  4 
times  3  sevenths  ? 

28.  Cornell  gave  3  times  3  sixths  of  an  apple  to  Gray, 
and  had  4  times  as  much  remaining ;  how  much  had  he 
at  first? 

29.  How  much  is  5  times  3  fourths,  minus  2  time?  3 
fourths,  plus  6  times  3  fourths,  minus  7  times  3  fourths'/ 


MENTAL  ARITHMETIC.  29 

30.  I  gave  A  6  tenths  of  a  dollar,  and  I  gave  B  3 
times  as  much,  plus  2  tenths  of  a  dollar ;  how  much  did 
I  give  both  ? 

31  What  will  1  fifth  of  a  yard  of  tape  cost,  at  the  rate 
of  20  fourths  cents  a  yard  ?  What  will  3  fifths  cost,  at 
the  same  rate  ? 

32.  Johnston  gave  his  brother  1  third  of  6  eighths  of  * 
part  of  walnuts ;  what  part  of  a  quart  did  he  receive  ? 

33.  How  much  is  1  fourth  of  8  sevenths?  of  12  ninths? 
of  16  tenths?  of  24  twelfths? 

34.  Stanton  having  2  thirds  of  a  dollar,  found  1  half 
of  4  thirds  of  a  dollar ;  how  many  thirds  had  he  then  ? 

35.  Mr.  A  bought  7  tenths  of  a  barrel  of  sugar,  and 
then  sold  2  thirds  of  6  tenths  of  a  barrel ;  how  much  re- 
mained? 

36.  What  is  the  difference  between  5  times  3  sevenths, 
and  1  fifth  of  40  sevenths  ? 

37.  Thornton  having  4  times  2  sixths  of  a  bushel  of 
corn,  bought  3  fourths  of  20  sixths  of  a  bushel;  how 
much  had  he  then  ? 

38.  If  1  half  of  a  yard  of  tape  cost  2  tenths  of  a  dime, 
how  many  yards  may  be  bought  fur  3  fourths  of  16  tenths 
of  a  dim  e  ? 

39.  Martin  sold  2  thirds  of  6  sevenths  of  a  peck  of 
beans,  ard  then  had  3  fourths  of  8  sevenths  of  a  peck 
remaining ;  how  many  had  he  at  first  ? 

40.  Ferris  lost  3  fourths  of  8  ninths  of  a  dollar,  and 
then,  having  found  3  ninths  of  a  dollar,  had  3  fourths  of 
8  ninths  remaining ;  how  much  had  he  at  first  ? 


LESSON  V. 

1    What  will  4  apples  cost,  if  3  apples  cost  9  cents  ? 

SOLUTION. — If  3  apples  cost  9  cents,  1  apple  will  cost  one 
third  of  9  cents,  which  is  3  cents;  and  if  1  apple  cost  3  cents,  4 
apples  will  cost  4  times  3  cenis,  whish  are  12  cents.  There- 
fore, &c. 


30  MENTAL  ARITHMETIC. 

2.  What  will  5  lemons  cost,  at  the  rate  of  3  for  12 
cents  ? 

3.  If  3  pairs  of  shoes  cost  6  dollars,  how  much  will  5 
pairs  cost? 

4.  What  will  9  candies  cost,  if  4  candies  cost  3  cents? 

5.  If  4  peaches  are  worth  8  cents,  what  are  8  peaches 
worth  ? 

6.  What  are  10  oranges  worth,  if  8  oranges  cost  10 
cents  ? 

7*  If  7  pounds  of  meat  cost  42  cents,  what  will  9 
pounds  cost? 

8.  What  cost  7  sheep,  at  the  rate  of  5  for  30  dollars? 

9.  What  cost  11  barrels  of  flour,  at  the  rate  of  5  barrels 
for  30  dollars  ? 

10.  If  12  boxes  of  figs  cost  48  dollars,  what  will  7 
boxes  cost? 

11.  How  much  will  7  cows  cost,  if  3  cows  are  bought 
for  60  dollars? 

12.  How  far  will  a  man  travel  in  12  days,  at  the  rate 
of  36  miles  in  4  days? 

13.  How  many  tons  of  hay  will  a  drover  feed  in  11 
weeks,  at  the  rate  of  10  tons  in  5  weeks  ? 

14.  How  much  must  be  paid  for  the  keeping  of  IS 
horses,  at  the  rate  of  80  cents  for  4  horses  ? 

15.  Required  the  value  of  5  ducks,  at  the  rate  of  120 
cents  for  3  ducks. 

16.  Mary  gave  10  cents  for  apples,  at  the  rate  of  3 
cents  for  9 ;  how  many  did  she  buy  ? 

17.  Fanny  paid  8  dollars  for  silk,  at  the  rate  of  5  dol- 
tars  for  15  yards;  how  many  did  she  buy? 

18.  Wilkinson  walked  7  hours,  at  the  rate  of  12  miles 
in  4  hours ;  how  far  did  he  travel  ? 

19.  Robert  gave  12  oranges  for  apples,  at  the  rate  of 
3  oranges  for  9  apples;  how  many  apples  did  he  get? 

20.  At  the  rate  of  3  melons  for  12  oranges,  how  many 
oranges  can  be  bought  for  10  melons? 

21.  If  6  men  can  mow  12  acres  of  grass  in  a  day,  how 
much  can  8  men  mow  in  the  same  time? 


MENTAL  ARITHMETIC.  31 

22.  If  1C  men  can  dig  30  rods  of  ditch  in  one  day, 
how  much  can  12  men  do  m  the  same  time? 

23.  How  long  will  it  take  4  ladies  to  drink  a  box  of 
tea,  if  6  ladies  can  drink  it  in  12  days? 

24.  If  5  boys  can  do  a  piece  of  work  in  16  days,  ho\v 
long  will  it  take  20  boys  to  do  it? 

25.  In  what  time  will  8  girls  pick  a  bushel  of  berries, 
if  4  girls  can  do  it  in  8  hours  ? 

26.  How  many  men  will  be  required  to  ouild  a  boat  in 
6  days,  if  3  men  can  do  it  in  12  days? 

27.  How  many  men  can  do  as  much  work  in  4  days, 
as  8  men  can  in  40  days? 

28.  If  it  require  10  men  8  days  to  build  a  wall,  how 
many  men  will  be  required  to  build  it  in  5  days  ? 

29.  If  5  men  build  a  boat  in  20  days,  how  many  men 
will  be  required  to  do  it  in  1  fourth  of  the  time  ? 

30.  What  cost  1  half  of  12  yards  of  cloth,  at  the  rate 
of  12  dollars  for  4  yards? 

31.  If  7  yards  of  cashmere  cost  21  dollars,  what  will  2 
thirds  of  15  yards  cost  ? 

32.  What  cost  3  fourths  of  8  pounds  of  coffee,  at  the 
rate  of  10  pounds  for  60  cents? 

33.  If  3  fifths  of  10  yards  of  ribbon  cost  30  cents, 
what  will  4  sixths  of  12  yards  cost? 

34.  Mary  gave  7  apples  for  21  chestnuts ;  at  this  rate 
how  many  chestnuts  could  she  get  for  8  apples? 

35.  If  8  lemons  are  worth  16  oranges,  how  many 
oranges  can  you  buy  for  10  lemons  ? 

36    At  the  rate  of  6  citrons  for  18  melons,  how  many 
melons  may  be  purchased  for  11  citrons  ? 

37.  If  9  apples  are  worth  27  chestnuts,  how  many 
chestnuts  may  be  had  for  12  apples? 

38.  If  3  sevenths  of  14  bunches  of  grapes  cost  24 
cents,  what  are  2  fifths  of  15  bunches  worth? 

39.  Peter  can  walk  3  fourths  of  8  miles  while  John 
walks  12 ;  how  far  can  John  go  while  Peter  walks  7  miles? 

40.  If  8  quarts  of  molasses  .cost  40  cents,  what  will  4 
«ixths  of  24  qua-ts  cost  ? 


32  MENTAL   ARITHMETIC. 

41.  What  must  I  pay  to  ride  1  half  of  14  miles,  if  it 
cost  me  20  dimes  to  ride  2  thirds  of  15  miles? 

42.  I  gave  8  yards  of  muslin  for  6  gallons  of  molasses; 
what  did  the  molasses  cost  a  gallon,  if  4  yards  of  muslin 
cost  48  cents  ? 

43.  A  gave  9  bushels  of  wheat  for  3  barrels  of  flour ; 
what  was  the  wheat  worth  a  bushel,  if  8  barrels  of  flour 
cost  72  dollars  ? 

44.  B  bought  7  yards  of  cloth  for  21  dollars,  and  gave 
4  yards  for  apples  worth  2  dollars  per  barrel;  how  many 
barrels  of  apples  did  he  receive  ? 

45.  C  gave  7  grammars  for  6  arithmetics ;  how  much 
were  the  grammars  worth  each,  if  5  arithmetics  cost  35 
dimes  ? 

46.  If  7  apples  cost  21  cents,  how  many  apples  must 
be  given  for  9  peaches,  bought  at  the  rate  of  3  for  12 
cents  ? 

47.  Reuben  had  9  oranges  worth  4  cents  apiece,  and 
Jackson  had  8  lemons  worth  3  cents  each,  which  he  gave 
to  Reuben  for  a  part  of  his  oranges ;  how  many  oranges 
had  Reuben  remaining  ? 


LESSON  VI. 

1.  What  will  one  yard  of  tape  cost,  if  2  thirds  of  a 
yard  cost  4  cents  ? 

SOLUTION. — If  2  thirds  of  a  yard  of  tape  cost  4  cents,  1  third 
of  a  yard  will  cost  1  half  of  4  cents,  which  is  2  cents,  and  3 
thirds,  or  one  yard,  will  cost  3  times  2  cents,  which  are  6  cents. 
Therefore,  &c. 

2.  What  will  one  box  of  soap  cost,  if  3  fourths  of  a 
box  cost  6  dollars  ? 

3.  If  4  fifths  of  a  box  of  tea  cost  8  dollars,  what  will 
one  box  cost  ? 

4.  If  3  fifths  of  a  yard  of  cloth  cost  6  dollars,  what 
will  one  yard  cost  ? 


MENTAL  ARITHMETIC.  33 

5.  What  will  2  pounds  of  starch  cost,  if  5  sixths  of  a 
pound  cost  10  cents? 

6.  What  cost  2  barrels  of  flour,  at  the  rate  of  4  dol- 
lars for  4  sixths  of  a  barrel  ? 

7.  If  7  eighths  of  a  keg  of  oysters  cost  14  shillings, 
what  will  be  the  cost  of  3  kegs  of  oysters  ? 

8.  What  cost  3  yards  of  cloth,  if  3  sevenths  of  a  yard 
cost  6  dollars  ? 

9.  How  far  can  A  walk  in  4  days,  if  in  5  sixths  of  a 
day  he  can  walk  20  miles  ? 

10.  What  cost  5  boxes  of  raisins,  if  3  fifths  of  a  box 
cost  6  dollars  ? 

11.  What  is  5  times  the  distance  to  Lancaster,  if  3 
fourths  of  the  distance  is  3  miles  ? 

12.  How  much  will  4  bushels  of  apples  cost,  if  5  tenths 
of  a  bushel  cost  50  cents? 

13.  What  cost  8  barrels  of  flour,  if  7  eighths  of  a  barrel 
cost  7  dollars? 

14.  Mary  bought  9  pecks  of  beans,  at  the  rate  of  12 
cents  for  6  sevenths  of  a  peck;  required  the  cost. 

15.  How  much  will  5  tons  of  hay  cost,  if  10  dollars 
will  buy  5  sixths  of  a  ton*? 

16.  What  is  the  cost  of  10  barrels  of  sugar,  at  the  rate 
of  20  dollars  for  4  fifths  of  a  barrel  ? 

17.  Pelton  bought  4  dozen  eggs  at  the  rate  of  8  cents 
for  2  thirds  of  a  dozen ;  how  much  did  they  cost  ? 

18.  What  is  the  value  of  1  half  of  6  bushels  of  peaches, 
at  the  rate  of  2  dollars  for  2  thirds  of  a  bushel  ? 

19.  What  cost  2  thirds  of  9  yards  of  cloth,  if  3  fourths 
of  twelve  yards  cost  27  dollars  ? 

20.  If  4  fifths  of  10  lemons  cost  24  cents,  what  cost  3 
fourths  of  12  lemons  ? 

21.  If  3  fourths  of  a  barrel  of  flour  cost  6  dollars,  what 
will  5  eighths  of  a  barrel  cost  ? 

22.  3  fourths  of  40  dollars  is  6  times  what  a  farmer 
gave  for  a  plough ;  required  its  cost. 

23.  A  watch  cost  30  dollars,  and  4  fifths  of  its  cost  is 
twice  the  cost  of  the  chain ;  what  was  the  cost  of  the  chain  ?' 


84  MENTAL  ARITHMETIC. 

24  B's  horse  cost  200  dollars,  and  3  fifths  of  its  cost 
lacks  80  dollars  of  being  4  times  the  cost  of  the  sleigh  j 
required  the  cost  of  the  sleigh. 

25.  A   merchant  having  20  barrels  of  flour,  sold   8 
fourths  of  it  to  A,  and  3  fifths  of  the  remainder  to  B ; 
bow  much  remained  ? 

26.  Elmina  is  25  years  old,  and  4  fifths  of  her  age  iy 
4  years  less  than  twice  Elmira's  age;  required  Elmira's 
age. 

27.  Think  of  a  number,  multiply  it  by  8,  divide  by  4, 
multiply  by  3,  divide  by  6,  add  20,  subtract  the  number 
thought  of,  divide  by  4,  and  name  the  result. 

28.  Think  of  a  number,  multiply  by  12,  divide  by  3, 
multiply  by  2,  divide  by  8,  add  12,  subtract  the  number 
thought  of,  divide  by  4,  and  name  the  result. 

29.  Multiply  10  by  12,  divide  by  6,  multiply  by  5, 
divide  by  2,  multiply  by  4,  divide  by  10,  and  name  the 
result. 

30.  How  many  are  24  multiplied  by  6,  divided  by  3, 
multiplied  by  8,  divided  by  4;  multiplied  by  2;  divided 
by  4;  divided  by  8  ? 


LESSON  VIL 

1.  How  many  thirds  are  there  in  4  ? 

SOLUTION. — In  1  there  are  3  thirds,  and  in  4  there  are  4  times 
8  thirds,  which  are  12  thirds.     Therefore,  £c. 

2.  How  many  thirds  in  2?  3?  5?  7?  8? 
8.  How  many  fourths  in  3?  5?  6?  4?  7? 

4.  How  many  fifths  in  5?  4?  3?  2?  8? 

5.  How  many  sixths  in  3?  2?  5?  6?  4? 

6.  How  many  sevenths  in  2?  5?  4?  7?  9? 

7  How  many  eighths  in  3?  6?  4?  5?  7? 

8  How  many  ninths  in  8?  4?  6?  3?  10? 


MENTAL  ARITHMETIC,  35 

*  9.  How  many  thirds  in  3   and  2  thirds?     In  4  and 
1  third? 

10.  How  many  fifths  in  4  and  3  fifths?     In  6  and  3 
fifths? 

11.  How  many  fourths  in  2  and  1  fourth?     In  7  and 
3  fourths  ? 

12.  How  many  sevenths  in  5  and  6  sevenths?     In  3 
and  4  sevenths? 

13.  How  many  ninths  in  3  and  4  ninths?     In  6  and 
7  ninths? 

14.  How  many  sixths  in  7  and  5  sixths?     In  3  and  2 
sixths  ? 

15.  How  many  eighths  in  5  and  3  eighths?    In  5  and 
7  eighths  ? 

16.  If  5  yards  of  cloth  cost  2  and  1  half  dollars,  wha* 
will  6  yards  cost? 

17.  If  4  pears  are  worth  2  and  2  third  cents,  what  are 
7  pears  worth? 

18.  What  cost  10  peaches,  at  the  rate  of  4  and  1  half 
cents  for  3  ? 

19.  If  11  ducks  cost  4  and  2  fifths  dollars,  what  will 
12  ducks  cost? 

20.  If  1  half  of  eight  yards  of  cloth  cost  3  and  1  fifth 
dollars,  what  cost  3  fifths  of  15  yards? 

21.  If  2  thirds  of  9  apples  cost  4  and  4  fifths  cents, 
what  will  3  fourths  of  12  apples  cost? 

22.  If  4  fifths  of  ten  pounds  of  sugar  cost  5  and  1  third 
dimes,  what  cost  5  sevenths  of  14  pounds  ? 

23.  What  cost  4  fifths  of  15  yards  of  muslin,  at  tho 
rate  of  7  and  1  half  dimes  for  3  fourths  of  20  yards  ? 

24.  How  many  whole  ones  in  6  thirds  ? 

SOLUTION. — In  one  there  are  three  thirds  ;  hence,  in  six  thirds 
there  are  as  many  ones  as  3  is  contained  times  in  6,  which  are 
2.  Therefore,  in  6  thirds  there  are  2  ones. 

25.  How  many  ones  in  6  halves?  9  thirds?  12  thirds? 

26.  How  many  ones  in  12  fourths  ?     20  fourths  ?     8 
fourths  ? 


36  MENTAL  ARITHMETIC. 

27.  How  many  ones   in   10  fifths?    12    sixths?   14 
sevenths  ? 

28.  How  many  ones  in  16  eighths?  21  sevenths?  24 
eighths  ? 

29.  How  many  ones  in  18   ninths?  15  thirds?   25 
fifths? 

30.  How  many  ones  in  28  sevenths?  36  ninths?  24 
fourths  ? 

31.  How  many  ones  in  15  thirds  ?  20  tenths  ?  33  elev- 
enths ? 

32.  How  many  ones  in  48  twelfths?  18  halves?  40 
tenths  ? 

33.  How  many  ones  in  9  halves?  7  thirds?  11  fourths? 

34.  How  many  ones  in  7  fifths?  16  fifths?  19  eighths  ? 

35.  How  many  ones  in  20   ninths?    17  sixths?    15 
fourths  ? 

36.  How  many  ones  in  13  fifths?  11  thirds?  17  fifths  ? 

37.  How  many  ones  in  25  eighths  ?  23  tenths  ?  31 
ninths  ? 

38.  If  2  apples  cost  6  fifths  of  a  cent,  what  cost  5 
apples  ? 

39.  If  3  pens  cost  9  eighths  of  a  cent,  what  cost  8 
pens? 

40.  If  5  pigs  cost  10  sevenths  of  a  dollar,  what  cost 
7  pigs? 

41.  What  cost  6  pairs  of  boots,  if  4  pairs  cost  12  fifths 
dollars  ? 

42.  What  are  8  pies  worth,  if  3  pies  are  worth  15 
fourths  cents? 

43.  What  cost  12  pine  apples,  if  3  cost  3  fourths  of  a 
dollar  ? 

44.  If  2  books  cost  4  fifths  of  a  dollar,  what  cost  10 
books? 

45.  How  much  are  9  lamps  worth,  if  5  are  worth  19 
thirds  dollars  ? 

46.  Required  the  cost  of  8  hats,  if  6  cost  12  fourths 
of  a  dollar. 


MENTAL  ARITHMETIC.  37 

47.  If  3  peaches  cost  9  eighths  of  a  cent,  what  cost  8 
peaches  ? 

43.  How  much  are  4  mirrors  worth,  if  7  are  worth  14 
halves  dollars? 

49  What  cost  3  halves  of  a  yard  of  linen,  if  5  yards 
cost  10  ninths  of  a  dollar  ? 

50.  What  cost  1  half  of  8  yards  of  cloth,  if  1  half  of 
6  yards  cost  1  third  of  27  fourths  dollars  ? 


LESSON  VIH. 

1.  3  is  1  half  of  what  number? 

SOLUTION. — If  1  half  of  some  number  is  3,  two  halves,  or  that 
number,  is  2  times  3,  which  are  6.     Therefore,  &c. 

2.  4  is  1  third  of  what  number? 

3.  6  is  1  fourth  of  what  number? 

4.  5  is  1  sixth  of  what  number? 

5.  6  is  1  half  of  what  number? 

6.  8  is  1  seventh  of  what  number? 

7.  9  is  1  fifth  of  what  number? 

8.  7  is  1  ninth  of  what  number? 

9.  4  is  1  fifth  of  what  number? 

10.  5  is  1  seventh  of  what  number? 

11.  10  is  1  sixth  of  what  number? 

12.  9  is  1  third  of  what  number? 

13.  11  is  1  fourth  of  what  number? 

14.  15  is  1  third  of  what  number? 

,      15.  A  is  10  years  old,  which  is  1  fifth  of  B's  age  ; 
required  B's  age. 

16.  An  apple  cost  3  cents,  which  is  1  fourth  of  the 
oost  of  a  melon  ]  required  the  cost  of  the  melon. 

17    A  sheep  cost  6  dollars,  which  is   1  fifth  of  the 
oost  of  a  cow ;  what  was  the  cost  of  the  cow  ? 
4* 


38  MENTAL  ARITHMETIC. 

18.  John  has  20  marbles,  which  is  1  third  of  Henry's 
number;  how  many  has  Henry? 

19.  Mary's  shawl  cost  7  dollars,  which  is  1   fourth 
of  the  cost  of  her  dress ;  required  the  cost  of  her  dress. 

20.  Henry  found   5   marbles,    which  is   1   third   of 
what  he  had ;  how  many  had  he  then  ? 

21.  6  is  1  half  of  3  times  what  number? 

22.  5  is  1  fourth  of  2  times  what  number?  . 

23.  8  is  1  third  of  4  times  what  number? 

24.  10  is  1  fourth  of  8  times  what  number  ? 

25.  9  is  1  half  of  6  times  what  number? 

26.  7  is  1  sixth  of  3  times  what  number? 

27.  12  is  1  third  of  9  times  what  number? 

28.  4  is  1  ninth  of  6  times  what  number  ? 

29.  11  is  1  sixth  of  3  times  what  number? 

30.  Flora's  cloak  cost  10  dollars,  which  is  1  third  of 
6  times  the  cost  of  her  dress ;  required  the  cost  of  hei 
dress. 

31.  A  watch  was  bought  for  20  dollars,    which  is  1 
third  of  4  times  what  the  chain  cost;  required  the  cost 
of  the  chain. 

32.  The  head  of  a  fish  is  6  inches  long,  which  is  1 
fou  th  of  3  times  the  length  of  the  body;  what  is  the 
length  of  the  body  ? 

33.  A.  slate  cost  20  cents,  which  is  1  fifth  of  4  times 
the  cost  of  an  arithmetic ;  required  the  cost  of  them  both. 

34.  A  boy  lost  15  cents, ,  which  is  1  fourth  of  5  times 
the  money  he  had  remaining ;  how  much  money  had  he 
at  first  ? 

35  Mary  found  12  pins,    which  is  1  fifth  of  3  times 
what  she  then  had;  how  many  had  she  at  first? 

86.  8  is  1  third  of  1  half  of  what  number  ? 

37.  4  is  1  fifth  of  1  third  of  what  number  ? 

38  3  is  1  fourth  of  1  fifth  of  what  number? 

39.  2  is  1  eighth  of  1  fourth  of  what  number? 

40.  3  is  1  seventh  of  1  sixth  of  what  number  ? 

41.  4  is  1  tenth  of  1  fifth  of  what  number  ? 
42  5  is  1  half  o^  1  seventh  of  what  number  ? 


MENTAL  ARITHMETIC.  39 

43.  6  is  1  fifth  of  1  third  of  what  nurnbei  f 

44    Philip's  vest  cost  4  dollars,  which  is  1  lialf  of  1 

third  of  the  cost  of  his  coat;  what  was  the  cost  of  his 

coat? 

45.  Martin  is  4  years  old,  and  his  age  is  1  third  of  1 
fourth  of  his  father's  age ;  how  old  is  his  father  ? 

46.  Ella's  bonnet  cost  3  dollars,  which  is  1  fifth  of  1 
half  of  the  cost  of  her  shawl ;  required  the  cost  of  the 
shawl 

47.  A  paid  10  dollars  for  a  saddle,  which  is  1  fifth 
of  1  third  of  the  cost  of  his  horse ;  required  the  cost  of 
the  horse. 

48.  Mr.  A  walked  8  miles,  which  is  4  times  1  seventh 
of  the  distance  he  rode ;  how  far  did  he  travel  ? 

49.  Benton's  house  cost  2000  dollars,   which  is  1   half 
of  4  times  the  cost  of  his  barn  ;  required  the  cost  of  both. 

50.  The  head  of  a  fish  is  3  inches  long,  and  the  tail  5 
inches,  which  is  1  half  of  1  third  of  the  length  of  the 
body ;  required  the  length  of  the  fish. 


LESSON  IX. 

1.  6  is  2  thirds  of  what  number? 

SOLUTION. — If  2  thirds  of  some  number  is  6,  1  third  of  that 
number  is  1  half  of  6,  which  is  3,  and  3  thirds,  or  that 
number,  is  3  times  3,  or  9.  Therefore,  &c. 

2.  9  is  3  fourths  of  what  number? 

3.  6  is  2  thirds  of  what  number? 

4.  10  is  2  fifths  of  what  number  ? 
5  12  is  4  sixths  of  what  number  ? 

6.  10  is  5  sevenths  of  what  number? 

7.  8  is  4  ninths  of  what  number  ? 

8.  9  is  3  fifths  of  what  number? 

9.  15  is  5  sixths  of  what  number? 
10.  10  is  5  eighths  of  what  number? 


40  MENTAL  ARITHMETIC. 

11.  16  is  8  ninths  of  what  number? 

12.  14  is  7  thirds  of  what  number  ? 

13.  Frank  is  12  years  old,  and  his  age  is  3  fifths  of 
Fanny's  age ;  how  old  is  Fanny  ? 

14.  Augustus  gave  his  brother  10  peaches,  which  is 
2  thirds  of  all  he  had ;  how  many  had  he  ? 

15.  Hampton  lost  8  cents,  which  is  4  fifths  of  Lis 
money ;  how  much  had  he  remaining  ? 

16.  A  lady  found  12  dollars,   which  is  4  sixths  of 
what  money  she  then  had ;  how  much  had  she  at  first  ? 

17.  Lester  sold  a  cow  for  24  dollars,  which  is  6  fifths 
of  the  cost  of  the  cow ;  required  its  cost. 

18.  A  farmer  sold  a  colt  for  80  dollars,  and  thereby 
gained  1  fifth  of  the  cost  of  the  colt ;  required  the  cost, 

19.  Mr.  M  is  20  years  of  age,  and  4  fifths  of  his  age 
is  8  ninths  of  his  brother's  age ;  what  is  the  age  of  his 
brother  ? 

20.  Frank  lost  12  marbles,  which  is  2  fifths  of  what 
he  had  at  first ;  how  many  remained  ? 

21.  10  is  1  half  of  4  fifths  of  what  number? 

22.  12  is  1  third  of  6  sevenths  of  what  number? 

23.  9  is  1  fourth  of  4  fifths  of  what  number  ? 

24.  16  is  2  fifths  of  10  fourths  of  what  number  ? 

25.  4  is  2  fifths  of  5  sixths  of  what  number  ? 

26.  6  is  3  fourths  of  4  fifths  of  what  number  ? 

27.  15  is  5  sixths  of  6  sevenths  of  what  number? 

28.  14  is  7  fourths  of  4  thirds  of  what  number  ? 

29.  18  is  9  eighths  of  4  sevenths  of  what  numbei  ? 

30.  20  is  5  fourths  of  8  thirds  of  what  number? 

81.  Thomas  sold  a  book  for  40  cents,  which  is  4 
fifths  of  5  sixths  of  the  cost  j  required  the  zost. 

32  Smiley  sold  his  horse  for  140  dollars,  which  is  7 
eighths  of  4  thirds  of  its  value ;  required  its  value. 

33.  A' s  hat  cost  6  dollars,    which  is  3  fourths  of  4 
fifths  of  the  cost  of  his  vest ;  required  the  cost  of  his  vest. 

34.  20  feet  of  a  pole  is  in  the  water,  which  is  2  fifths 
of  5  sevenths  of  the  length  in  the  air;  what  is  the  length 
of  the  pole  ? 


MENTAL  ARITHMETIC.  41 

35  A  pole  is  30  feet  in  the  air,  which  is  3  fifths  of  2 
fourths  of  the  length  of  the  pole ;  required  the  length  in 
the  mud  and  water. 

36.  A  cow  cost  24  dollars,  which  is  6  tenths  of  2  fifths 
of  the  cost  of  the  cow  and  a  horse ;  what  was  the  cost  of 
the  horse  ? 

37.  A  man  sold  his  watch  for  60  dollars,  which  is   5 
fourths  of  4  times  what  the  chain  cost,  and  the  watch 
cost  3  times  as  much  as  the  chain ;  what  was  the  cost  of 
each  ? 

38.  A's  horse  cost  200  dollars,  and  4  fifths  of  the  cost 
f  the  horse  is  8  times  the  cost  of  his  harness }  required 

the  cost  of  the  harness. 

39.  A  man  has  24  geese,  and  3  fourths  uf  the  numbei 
of  geese  equals  2  times  the  number  of  turkeys;  how 
many  turkeys  had  he  ? 

40.  A  man  sold  his  horse  and  sleigh  for  200  dollars, 
and  4  fifths  of  this  is  8  times  what  his  sleigh  cost,  and 
the  horse  cost  10  times  as  much  as  the  sleigh ;  required 
the  cost  of  each. 


LESSON  X. 

1.  6  are  how  many  times  1?  2?  3? 

2.  8  are  how  many  times  1?  2?  4? 

3.  12  are  how  many  times  2?  3?  6? 

4.  20  are  how  many  times  4?  5?  10? 

5.  32  are  how  many  times  4?  8?  16? 

6.  30  are  how  many  times  5?  6?  10? 

7.  72  are  how  many  times  6?  8?  9? 
8    80  are  how  many  times  4?  8?  20? 
9.  What  is  the  relation  of  8  to  2? 

REMARK  —8  is  4  times  2. 

10.  What  is  the  relation  of  12  to  4?     Of  15  to  5? 

11  What  is  the  relation  of  16  to  8  ?     Of  18  to  6? 

12  What  is  the  relation  of  21  to  7  ?     Of  24  to  8  ? 


42  MENTAL  ARITHMETIC. 

13  What  is  the  relation  of  32  to  4  ?     Of  27  to  9  ? 

14  What  is  the  relation  of  40  to  10?     Of  45  to  5? 

15.  What  is  the  relation  of  28  to  7?     Of  36  to  4? 

16.  If  4  yards  of  cloth  cost  10  dollars,  what  will  8 
yards  cost  ? 

SOLUTION. — If  4  yards  of  cloth  cost  10  dollars,  8  yards,  whicn 
are  2  times  4  yards,  will  cost  2  times  10  dollars,  or  20  dollars. 
Therefore,  &c. 

17.  If  3  bunches  of  grapes  cost  8  cents,  what  will  6 
bunches  cost  ? 

18.  If  6  combs  cost  9  cents,  what  will  12  combs  cost 
at  the  same  rate  ? 

19.  If  7  peaches  cost  8  cents,  what  will  21  peaches 
cost? 

20.  If  5  pairs  of  shoes  cost  9  dollars,  what  will  20  pains 
cost? 

21.  If  4  pens  cost  11  cents,  what  will  12  pens  cost? 

22.  If  8  ducks  cost  5  dollars,  what  will  24  ducks  cost? 

23.  What  cost  30  lamps,  if  5  lamps  cost  7  dollars  ? 

24.  How  much  will  42  primers  cost,  at  the  rate  of  6 
primers  for  20  cents  ? 

25.  What  cost  56  inkstands,  if  7  inkstands  cost  5  dol- 
lars? 

26.  If  5  pitchers  cost  3  dollars,  what  will  45  pitchers 
cost? 

27.  How  far  will  a  man  travel  in  48  days,  if  he  travel 
30  miles  in  4  days  ? 

28.  If  6  men  can  build  10  rods  of  wall  in  a  certain 
tame,  how  many  rods  can  54  men  build  in  the  same  time? 

29.  Hiram  bought  6  pigs  for  11  dollars;  how  many 
could  he  have  bought  for  44  dollars  ? 

30.  7  men  earn  12  dollars  in  3  days;  how  much  could 
they  earn  in  27  days  ? 

31.  If  5  peaches  are  worth  one  pear,  how  many  pears 
are  30  peaches  worth  ? 

32.  If  8  dollars  will  buy  5  gold  pens,  how  many  wiU 
56  dollars  buy  ? 


MENTAL  ARITHMETIC.  43 

33.  If  6  stands  cost  2  thirds  of  12  dollars,  what  will 
30  stands  cost? 

34.  40  dollars  is  2  thirds  of  what  A  gave  for  sheep, 
at  the  rate  of  10  dollars  for  3  sheep;  how  many  did  he 
purchase  ?  , 

35.  If  4  men  can  perform  a  piece  of  work  in  18  days, 
how  long  will  it  require  12  men  to  do  it  ? 

36.  If  6  men  can  build  a  boat  in  10  fourths  days,  how 
long  will  it  take  3  men  to  build  it  ? 

37.  15  dollars  is  3  eighths  of  what  A  earns  in  5  days  j 
how  much  will  he  earn  in  15  days  ? 

38.  18  men  are  3  fifths  of  the  number  required  to  mow 
a  field  in  8  days ;  how  many  men  would  be  required  to 
mow  it  in  24  days  ? 

39.  20  dollars  is  4  dollars  more  than  2  thirds  of  4 
times  what  B  paid  for  a  chain,  and  his  watch  cost  5  timee 
as  much  as  the  chain ;  required  the  cost  of  each  ? 


LESSON  XL 

1.  4  is  what  part  of  8  ? 

SOLUTION.— 4  is  1  half  of  8,  since  2  times  4  are  8. 

2.  3  and  6  are  what  parts  of  12  ? 

3.  4  and  8  are  what  parts  of  16  ? 

4.  3  and  6  are  what  parts  of  24  ? 

5.  7  and  3  are  what  parts  of  21  ? 

6.  4  and  9  are  what  parts  of  36  ? 

7.  What  is  the  relation  of  2  to  6  ?     Of  4  to  8  ? 

8.  What  is  the  relation  of  3  to  9  ?     Of  5  to  10  ? 

9.  What  is  the  relation  of  3  to  12  ?     Of  4  to  16  ? 
10.  What  is  the  relation  of  5  to  20  ?     Of  6  to  36  ? 

11  What  is  the  relation  of  7  to  42  ?     Of  8  to  40  ? 

12  What  is  the  relation  of  6  to  54  ?     Of  9  to  27  ? 


44  MENTAL  ARITHMETIC. 

13.  What  is  the  relation  of  10  to  40?     Of  7  to  56? 

14.  What  is  the  relation  of  11  to  55?     Of  12  to  48? 

15.  If  6  apples  cost  10  cents,  what  wiL  3  apples  cost? 

REMARK. — 3  apples,  the  half  of  6,  will  cost  one-half  of  10 
cents. 

16.  How  much  will  5  books  cost,  if  20  books  cost  16 
dollars? 

17.  What  cost  3  knives,  if  18  knives  cost  24  dollars? 

18.  If  14  pencils  cost  35  cents,  what  will  2  pencils 
cost? 

19.  If  10  peaches  are  worth  12  oranges,  how  many 
oranges  are  5  peaches  worth  ? 

20.  How  much  will  4  apples  cost,  if  16  apples  cost  24 
cents  ? 

21.  How  much  will  9  pigs  cost,  if  27  pigs  cost  36 
dollars  ? 

22.  What  cost  7  tons  of  hay,  if  56  tons  cost  96 
eagles  ? 

23.  If  42  sheep  are  sold  for  108  dollars,  what  are  7 
sheep  sold  for? 

24.  If  100   pens  cost  50  cents,  what,  will  20  pens 
cost  ? 

25.  What  cost  5  inkstands,  if  15  inkstands  cost  2 
thirds  of  18  dimes  ? 

26.  If  5  eighths  of  32  hens  cost  10  dollars,  what  will 

2  thirds  of  6  hens  cost? 

27.  If  3  fourths  of  48  oranges  cost  40  cents,  what  will 

3  fourths  of  1 2  oranges  cost  ? 

28.  If  A  walked  132  miles  in  33  days,  how  far  did  he 
walk  in  3  days  ? 

29.  If  8  times  6  cents  is  48  cents,  how  much  is  2 
times  6  cents  ? 

30.  Mary  having  27  roses  gave  1  third  of  them  to 
Sallie  and  1  third  to  Annie  5  how  many  remained  ? 

31.  A  worked  5  weeks  for  7  dollars  a  week,  and  re- 
ceived in  payment  12  bushels  of  wheat  worth  1  dollar 
and  1  half  a  bushel;  how  much  remains  due  A? 


MENTAL  ARITHMETIC.  45 

32    A  farmer  gave  2  thirds  of  15  bushels  of  rye,  worth 

dimes  a  bushel,  for  cloth  worth  3  dollars  a  }ard  ;  how 
many  yards  did  he  receive  ? 

33.  Think  of  a  number,  multiply  by  10,  divide  by  5, 
maltiply  by  3,  divide  by  6,  add  30,  subtract  the  original 
number,  divide  by  10,  add  7,  and  the  result  is  what  ? 

3  .  What  is  the  value  of  27  multiplied  by  8,  divided 
by  4,  multiplied  by  6,  divided  by  3,  multiplied  by  9, 
divided  by  12  ? 


LESSON  XII. 

1.  What  numbers  multiplied  together  will  produce  4? 
8?  10?  16?  12?  18?  24? 

2.  What  numbers  multiplied  together  will  produce  15? 
21?  28?  35?  36?  39?  48? 

3.  What  numbers  multiplied  together  will  produce  40? 
42?  45?  49?  50?  51?  52? 

Numbers  which  can  be  produced  by  multiplying  together  other 
numbers,  each  of  which  is  greater  than  a  unit,  are  called  com- 
posite numbers. 

Numbers  which  cannot  be  produced  by  multiplying  together 
two  or  more  numbers,  each  of  which  is  greater  than  a  unit,  are 
called  prime  numbers. 

4.  Tell  which  of  the  following  numbers  are  prime,  and 
which  composite;  4,  5,  6,  7,  8,  9,  10, 11, 12,  13, 14, 15, 
16,  17,  18,  19,  20,  21,  22,  23. 

5.  Name  the  prime  and  composite  numbers  in  the  fol- 
lowing list;  24,  25,  26,  27,  28,  29,  30,  31  32,  33,  34. 
85,  36,  37,  38,  39,  40,  41. 

The  numbers,  which  when  multiplied  together  will  produce  d 
composite  number,  are  called  factors  of  that  number. 

6    What  are  the  factors  of  12 1  20  ?  16  ?  33  ?  30  ?  24? 
27?  18?  26?  32? 
5 

.iVERSlTY 


16  -  MENTAL  ARITHMETIC 

7.  What  are  the  factors  of  9?  10?  14?  34  f  36?  40? 

48?  50?  56?  60?  63?  72? 

When  these  factors  are  prime  numbers,  they  are  called  prime 
factors. 

8.  What  are  the  prime  factors  of  4  ?  6  ?  9  ?  12  ?  15  ?  1 8  ? 

9.  What  are  the  prime  factors  of  10?  20?  21?  22 v 
24?  25? 

10.  What  are  the  prime  factors  of  27?  28?  30?  32? 
33?  35? 

11.  What  are  the  prime  factors  of  44?  45?  46?  48? 
49?  50? 

12.  What  are  the  prime  factors  of  52?  54?  55?  56? 
57?  60? 

13.  What  are  the  prime  factors  of  64?  68?  70?  72? 
75?  80? 

14.  What  prime  factors  are  common  to  6  and  12  ? 

15.  What  prime  factors  are  common  to  9  and  12  ? 

16.  What  prime  factors  are  common  to  8  and  20  ? 

17.  What  prime  factors  are  common  to  JO  and  20? 

18.  What  prime  factors  are  common  to  12  and  18? 

19.  What  prime  factors  are  common  to  16  and  24? 

20.  What  prime  factors  are  common  to  14  and  42  ? 

21.  Is  a  number  exactly  divisible  by  any  numbers  ex- 
cept its  prime  factors,  or  some  product  of  them  ? 

22.  Name  the  divisors  of  4,  6,  8, 10, 12, 14, 16,  and  20. 

23.  What  divisors  are  common  to  4  and  6?    To  8 
and  10? 

24.  What  divisors  are  common  to  6  and  9?     To  9 
and  18? 

25.  What  divisors  are  common  to  10  and  30?     To  * 
and  24? 

26.  What  divisors  are  common  to  9  and  27?     To  10 
and  20? 

27.  What  divisors  are  common  to  16  and  24?     To  12 
and  18? 

A  divisor  common  to  two  or  more   numbers  is  called  their 
common  divisor. 


MENTAL  ARITHMETIC.  47 

28  What  is  a  common  divisor  of  8  and  24  ?  Of  9 
and  15? 

29.  What  is  a  common  divisor  of  15  and  20?     Of  18 
and  80  ? 

30.  What  is  a  common  divisor  of  16  and  32  ?     Of  32 
and  40? 

The  greatest  divisor  common  to  two  or  more  numbers  is  called 
their  greatest  common  divisor. 

31.  What  is  the  greatest  common  divisor  of  4  and  8? 

32.  What  is  the  greatest  common  divisor  of  8  and  24? 

33.  What  is  the  greatest  common  divisor  of  3  aod  27? 
34    What  is  the  greatest  common  divisor  of  16  and  24  ? 

35,  What  is  the  greatest  common  divisor  of  24  and  32  ? 

A  multiple  of  a  number  is  any  number  that  will  contain  it,  a 
vshole  number  of  times,  without  a  remainder. 

36.  What  is  a  multiple  of  4?  Of  3?  Of  5?  Of  6?  Of  8? 

87.  What  is  a  multiple  of  7?     Of  9?     Of  10?     Of 

of  20? 

A  multiple  common  to  two  or  more  numbers  is  called  their 
common  multiple. 

88.  What  is  a  common  multiple  of  2  and  3  ?    Of  3  and 
4?     Of  4  and  5?     Of  5  and  6? 

39.  What  is  a  common  multiple  of  6  and  7?     Of  4 
and  6 '(     Of  5  and  10  ?     Of  9  and  12  ? 

Tlu  least  multiple,  common  to  two  or  more  numbers,  is  called 
their  k-w  common  multiple. 

40.  What  is  the  least  common  multiple  of  4  and  6? 
Of  6  and  S  ?     Of  8  aud  10  ?     Of  10  and  12  ? 

41  What  is  the  least  common  multiple  of  8  and  12? 
Of  9  and  6  >  Of  9  and  12?  Of  12  and  20? 

If  a  number  be  multiplied  by  itself,  the  result  is  called  the 
square  of  the  number;  if  the  square  be  multiplied  by  the  num- 
ber, the  result  is  the  cube,  it  the  cube  be  multiplied  by  the 
number,  the  result  is  the  fourth  power,  £c. 

42.  What  is  the  square,  cube,  and  fourth  power  of  Ijf 
2?  3?  4?  5?  6?  TV  S?  9?  10?  11?  12? 


48  MENTAL  ARITHMETIC. 

The  square  root  of  a  number  is  one  of  the  two  equal  factors 
which  produce  that  number;  the  cube  root,  one  of  the  throe 
equal  factors ;  the  fourth  root,  one  of  the  four  equal  factors. 

43.  What  is  the-square  root  of  1?  4?  9?  16?  25? 
49?  81?  36?  64? 

44  What  is  the  cube  root  of  1?  8?  27?  64?  125? 
313?  729?  512? 

45.  What  is  the  fourth  root  of  1?  16?  81  ?  256?  625? 
.46.  Define  a  prime  number,  composite  number,  factors, 
prime  factors,  common  divisor,  least  common  divisor,  com- 
mon multiple,  least  common  multiple,  square,  cube,  and 
fourth  power;  square,  cube,  and  fourth  roots. 


SECTION    III. 

LESSON  I. 

The  number  of  parts  into  which  anything  is  divided,  instead 
of  being  expressed  by  a  word,  may  be  represented  by  a  figure 
beneath  a  line,  thus: — 

U  represents  halves,  4-  represents  fourths, 

•5  represents  thirds,  g  represents  fifths. 

The  number  of  fractional  parts  taken  may  be  represented  by 
a  figure  above  the  line,  thus:  — 

|  represents  2  fourths,  f  represents  5  sevenths, 

I  represents  3  fifths,  J  represents  7  eighths, 

|  represents  4  sixths,  |  represents  4  ninths. 

These  expressions  are  called  fractions.  A  fraction  is  that 
definite  part  which  a  portion  is  of  the  whole.  Custom,  however, 
has  sanctioned  the  use  of  the  word  fraction  to  denote  both  the 
definite  part  and  the  expression  of  it.  The  expression  consists 
of  two  figures  with  a  line  between  them. 

The  figure  be".ow  the  line  is  the  denominator;  it  denotes  the 
number  of  equal  parts  into  which  the  unit  is  divided. 


MENTAL  ARITHMETIC.  49 

The  figure  above  the  line  is  the  numerator;  it  denotes  he 
number  of  equal  par, 3  taken. 

A  proper  fraction  is  one  whose  value  is  less  than  a  unit ;  ae 

I.I- 

An  improper  fraction  is  one  whose  value  is  equal  to,  cr  greater 
than  a  unit;  as  4  7 

A  mixed  number  consists  of  a  whole  number  and  a  traction  j 
as,  2i,  3|. 

The  reciprocal  of  a  number  is  a  unit  divided  by  that  number ; 
thus  the  reciprocal  of  4  is  1 ;  of  7  is  1. 

1.  How  many  halves  in  3^?  2|?  4^?  64? 

2.  How  many  thirds  in  3|?  2|?  4|  ?  5|? 

3.  How  many  fourths  in  3|  ?  4|  ?  2|  ?  7|  ? 

4.  How  many  fifths  in  11?  2|?  3|?  4|? 

5.  How  many  sixths  in  21?  3|?  4|?  5|? 

6.  How  many  eighths  in  2|?  61?  7|?  8|? 

7.  How  many  thirds  in  5|  ?  7|?  9|?  10|? 

8.  How  many  sevenths  in  3^?  51?  4|?  2^? 

9.  How  many  fifths  in  5|?  4J?  7|?  8|? 

10.  How  many  ninths  in       2|?     3|?     7|?     6|? 

11.  How  many  tenths  in       5T30?  7f<j?  3T«0?  6T70? 

12.  If  one  yard  of  tape  cost  21  cents,  how  many  third? 
cents  will  5  yards  cost? 

13.  How  many  fourths  of  a  dollar  will  7  baskets  of 
peaches  cost,  at  2^  dollars  a  basket  ? 

14.  What  cost  3  barrels  of  flour,  if  21  barrels  cost  18 
dollars  ? 

15.  How  far  can  Henry  walk  in  7  hours,  if  he  can 
walk  ]2*uiiles  in  2|  hours? 

16  If  3^  yards  of  muslin  cost  44  cents,  how  much 
will  10  yards  cost  ? 

17.  2 \  bushels  of  apples  cost  ^  dollars,  what  will  2J 
bnshels  cost  ? 

18.  How  many  ones  in  |?     |?        I*?     |? 

19.  How  many  ones  in  !„?     |9        u>?      12? 

20.  How  many  ones  in  f  ?     \4  ?      y  ?      y  ? 

21.  How  many  ones  in  |?      U  ?      lf  ?     21  ? 

22  How  many  ones  in  4  ?      if  ?      ^  ?     ||  ? 

23  How  man^  cnes  in  C5  ?   if"      lg?     y"? 
6* 


50  MENTAL  ARITHMETIC. 

24.  How  many  ones  in  y  ?     is  ?     u  ?     u>  ? 
25    How  many  ones  in  y*  ?     2/  ?     f*?     2^7? 

26.  How  many  ones  in  U  ?     2_*?     ||?     i|  ? 

27.  Reduce  3£,  7f ,  8-J,  and  2^  to  improper  fractions. 

28.  Reduce  y,  1/)  i^  an(j  2.3  to  mixed  numbers. 

29.  Reduce  5|,  6|,  7f ,  and  8|  to  improper  fractions. 

30.  Reduce  ~T4,  2H5,  ^6,  and  f  J  to  mixed  numbers. 

31.  Reduce  2|,  3|,  4T5Q,  and  5T<Y  to  improper  fractions, 

32.  At  |  of  a  dime  a  pound,  what  will  9  pounds  of 
candies  cost  ? 

83.  If  1  pound  of  sugar  cost  6|  cents,  what  will  8 
pounds  cost? 

SOL. — 8  pounds  will  cost  8  times  6f  cents.  8  times  6  is  48, 
8  times  f  is  ^  or  5J;  48  plus  5J  equals  53J. 

34.  At  6 1  dimes  a  bushel,  what  will  12  bushels  of 
wheat  cost  ? 

35.  What  cost  8  barrels  of  apples,  at  the  rate  of  3| 
dollars  a  barrel  ? 

36.  How  much  will  a  man  earn  in  a  week,  at  the  rate 
of  2 1  dollars  a  day? 

37.  At  the  rate  of  18|  cents  a  dozen,  what  will  3 
dozen  eggs  cost? 

38.  What  cost  5|  boxes  of  butter,  at  the  rate  of  3| 
dollars  for  10  boxes  ? 

39.  What  cost  7  barrels  of  apples,  at  the  rate  of  9 
dollars  for  2i  barrels  ? 

40.  How  many  chestnuts  must  be  given  for  12 1  cents, 
if  14  chestnuts  cost  3i  cents  ? 

41.  What  cost  5|  pounds  of  beef,  if  2  pounds  cost 
32  cents  ? 

42.  How  far  can  a  man  travel  in  5|  hours,  if  he  can 
walk  19  miles  in  2-^  hours  ? 

43.  A  vessel  sailed  23  miles  in  4|  hours;  how  far  did 
she  sail  in  2|  hours  ? 

44.  A  kite  arose  48  rods  in  3|  minutes;  how  far  at 
this  rate  will  it  ascend  in  3^  minutes  ? 

45.  What  cost  50  pounds  of  meat,  at  the  rate  of  4f 
pounds  for  5|  cents  ? 


MENTAL  ARITHMETIC.  5] 

46.  What  cost  10  yards  of  ribbon,  if  4J  yards  cost  6| 
cents  ? 

47.  If  f  of  a  pint  of  almonds  cost  |f  of  a  shilling, 
what  cost  5 1  pints  ? 

48.  If  4  of  a  ton  of  hay  is  worth  4|  dollars,  what  cost 
7 1  tons  of  hay  ? 

49.  If  a  m.m  walk  3  miles  in  2^  hours,  ho^w  far  will 
he  walk  in  ^  hours? 


LESSON  II 


SOLUTION.—  In  1  there  are  4,  and  iu  ^  there  are  |  of  |,  or  | 
Therefore,  &c. 


1.  How  many  fourths  in 

SOLUTION. — In  1  there  are  4,  t 
jrefore,  &c. 

2.  How  many  sixths  in            2?  '  1^  f^  1^ 

3.  How  many  eighths  in         <|?  |?  f?  |? 

4.  How  many  tenths  in           £?  ±?  f  ?  |? 

5.  How  many  twelfths  in        |  ?  ^  ?  £  ?  |  ? 

6.  How  many  fourteenths  in^?  |?  4^  f  ^ 

7.  How  many  fifteenths  in      |  ?  |?  |?  |? 

8.  How  many  sixteenths  in     |?  |?  §?  |? 

9.  How  many  eighteenths  in -|  ?  |?  |?  -|? 

10.  How  many  twentieths  in    |?     |?     y^?  |? 

11.  Reduce  |,  |,  and  |  to  twelfths. 

12.  Reduce  J,  |,  |,  and  T9Q  to  twentieths. 

13.  Reduce  ^,  4.  |,  T%  and  ji  to  thirtieths. 

14.  Reduce  |,  |,  T70-.  |?  and  ^§  to  fortieths. 

When  fractions  have  the  same  denominator   they  are  sai'-l  to 
have  a  common  denominator. 

15.  Reduce  |  and  |  to  a  common  denominator 

16.  Reduce  |  and  |  to  a  common  denominator. 

17.  Reduce  ^  and  i  to  a  common  denominator. 

18.  Reduce  i  and  |  to  a  common  denominator. 

19.  Reduce  |  arid  |  to  a  common  denominator. 

20.  Rsduce  i  and  I  to  a  common  denominator. 


62  MENTAL  ARITHMETIC. 

21  Reduce  |  and  f  to  a  common  denominator. 

22.  Reduce  §  and  |  to  a  common  denominator. 

23.  Reduce  |  and  |  to  a  common  denominator. 

24.  Reduce  T60  ami  |  to  a  common  denominator. 

25.  Reduce  |  and  |  to  a  common  denominator. 

26.  Reduce  f  and  T3^  to  a  common  denominator. 

27  Rqduce  i,  |,  and  \  to  a  common  denominator 

28  Reduce  |,  |,  and  J  to  a  common  denominator. 
29,  Reduce  ^,  ^,  and  ^  to  a  common  denominator. 
SO.  Reduce  |,  £,  and  y1^  to  a  common  denominator. 

The  five  following  question  are  designed  to  illustrate  the 
manner  in  which  the  pupil  may  be  led  to  derive,  by  induction, 
an  abbreviated  method  of  obtaining  the  same  results  as  those 
given  by  the  analysis.  Similar  questions  occur  in  several  lessons, 
and  claim  particular  attention. 

31.  Since  |  —  J,  by  what  number  may  you  multiply 
both  numerator  and  denominator  of  f  to  obtain  |  ? 

32.  Since  f  =  T92,  by  what  number  may  you  multiply 
both  numerator  and  denominator  of  |  to  obtain  T92  ? 

33.  Does  it  change  the  value  of  a  fraction  to  multiply 
both  numerator  and  denominator  by  the  same  number  ? 

34.  By  what  must  you  multiply  the   numerator  and 
denominator  of  |  to  reduce  it  to  tenths  ? 

35.  By  what  must  you  multiply  both  numerator  and 
denominator  of  f  to  reduce  it  to  twentieths  ? 


36.  Reduce 


38.  Reduce  f,  £,  j 
89.  Reduce  1,  %,  | 
40  Reduce 


2'    3>    6 


and  T80  to  twentieths, 
and  T70  to  thirtieths, 
and  |  to  twelfths. 
and  ^  to  eighteenths, 
and  I  to  sixteenths. 


41.  If  2^  yards  of  silk  cost  20  dimes,  what  will  4 
yards  cost? 

42.  If  2£  yards  of  tape  cost  13  cents,  what  will  3£ 
yards  cost? 

43.  What  cost  3£  pounds  of  sugar,  if  2\  pounds  cost 
25  cents  ? 

44.  Twice  18  is  |  of  a  certain  number;  required  the 
number. 


MENTAL   ARITHMETIC.  53 

45.  Mary  lost  20  roses,  which  is  |  as  many  as  she 
then  had;  how  many  had  she  at  first? 

46.  John  found  60  cents,  which  is  |  of  |  of  what  he 
then  had;  how  much  had  he  at  first? 

47.  If  |  of  |  of  a  yard  of  lace  cost  8  cents,  what  will 
|  of  9  yards  cost  ? 

48.  Henry  gave  his  sister  20   cents,  which  is  |  of 
what  he  had  at  first  more  than  his  sister,  and  ^  of  what 
she  now  has;  how  much  had  each  at  first? 


LESSON  III. 

1.  How  many  thirds  are  equal  to  |  ? 

SOLUTION. — 1  is  equal  to  |,  therefore   1   of  the  number  of 
sixths  equals  the  number  of  thirds  ;  1  of  4  is  2.  Therefore,  &. 

2.  How  many  halves  in 

3.  How  many  thirds  in 

4.  How  many  fourths  in 

5.  How  many  sixths  in 

6.  How  many  eighths  in 

7.  How  many  fifths  in 

8.  How  many  sevenths  in 

9.  How  many  ninths  in 

10.  How  many  tenths  in 

11.  Since  J  =  |,  by  what  may  we  divide  both  nume- 
rator and  denominator  of  |  to  produce  |  ? 

12.  Does  dividing  both  numerator  and  denominator  of 
a  fraction,  by  the  same  number,  change  its  value  ? 

13.  By  what  number  must  we  divide  both   numerate! 
and  denominator  of  |  to  reduce  it  to  fourths  ? 

14.  By  what  must  we  divide  both  numerator  and  de- 
nominator of  -,5Q  to  reduce  it  to, halves? 

15    Reduce  ffl  to  fifths,  and  T3^  to  fourths. 

16.  Reduce  ^  to  halves,  and  7y2  to  fourths. 

17.  Reduce   9j  to  fourths,  and  |  to  thirds. 


64  MENTAL  ARITHMETIC 

18.  Reduce  ^g  to  fifths,  and  ^4-  to  sixths. 

19.  Reduce  29T  to  sevenths,  and  ^|  to  ninths. 

20.  Reduce  -J|  to  eighths,  and  |'jj  to  tenths. 

21.  Reduce  ||  to  tenths,  and  ||  to  twelfths. 

22.  Reduce  f  f  to  ninths,  and  f  f  to  elevenths. 

When  a  fraction  cannot  be  reduced  to  an  equivalent  one  hav- 
ing a  less  denominator,  it  is  said  to  be  in  its  lowest  terms. 

23.  Reduce  |  and  T65  to  their  lowest  terms. 

24.  Feduce  {S  and  T9^  to  their  lowest  terms. 

25.  Reduce  ||  and  -f-%  to  their  lowest  terms. 
26    Reduce  £J  and  ||  to  their  lowest  terms. 

27.  Reduce  ||  arid  f  |  to  their  lowest  terms. 

28.  Reduce  |i  and  f  |  to  their  lowest  terms. 

29.  Reduce  ||  and  f  |  to  their  lowest  terms. 

30.  If  8  is  I  of  some  number,  what  is  |  of  the  same 
number  ? 

31.  If  6  is  |  of  some  number,  what  is  |  of  3  times  the 
same  number? 

32.  If  8  is  |  of  some  number,  what  is  |  of  2  times  the 
same  number? 

33.  Henry's  horse  cost  90  dollars,  which  is  T9a  of  5 
times  the  cost  of  the  sleigh;  required  the  cost  of  the 
sleigh. 

34.  4  times  50  years  is  10  years  less  than  10  times  the 
age  of  James j  how  old  is  he? 

35.  If  4  horses  eat  2  tons  of  hay  in  8  weeks,  aow  long 
will  it  require  5  horses  to  eat  the  same  ? 

36    If  8  men  can  build  a  boat  in  16  days,  how  long 
will  it  require  32  men  to  build  it  ? 

37.  How  many  lemons  will  pay  for  7  melons,  if   6 
ksinons  are  worth  4^  melons  ? 

38.  If  it  require  8|  yards  of  cloth  to  make  2  coats, 
how  much  will  be  required  to  make  9  coats  ? 

39.  42  dollars  is  f  of  all  the  money  A  has,  and  B  haa 
3  times  as  much ;  how  much  money  has  B  ? 

40.  What  number  being  multiplied  by  6,  divided  by  3, 
multiplied  by  5,  divided  by  2,  and  10  added,  equals  30? 

41 .  What  number  being  multiplied  by  8;  divided  by 


MENTAL  ARITHMETIC  55 

4,  multiplied  by  3,   divided  by  2,  and   10  subtracted, 
equals  14  ? 

42.  A -gave  B  48  cents,  and  |  of  this  is  4  time?,  as 
many  as  he  had  remaining;  how  much  had  he  at  first  ? 

43.  Amanda  having  50  pins,  lost  |  of  them,  and  theu 
found  |  as  many  as  remained ;  how  many  had  she  then  ? 

44.  A  watch  cost  $90,  which  is  |  of  10  times  what 
the  chain  cost;  required  the  cost  of  both. 

45.  Mary  gave  Lilly  24  pins,  which  is  f  of  what  Lilly 
already  had,  and  |  of  what  Mary  had  remaining;  how 
many  had  each  at  first? 


LESSON  IV. 

1.  What  is  the  sum  of  \  and  f  ? 

2.  What  is  the  sum  of  f  and  f  ? 

3.  What  is  the  sum  of  |  and  |  ? 

4.  What  is  the  sum  of  |  and  |  ? 

5.  What  is  the  difference  between  f  and  f  ? 
fi.  What  is  the  difference  between  |  and  |  ? 

7.  What  is  the  difference  between  |  and  |j  ? 

8.  What  is  the  difference  between  2|  and  If  1 

9.  What  is  the  difference  between  3|  and  2|i 

10.  How  many  fourths  in  -|  and  |  ? 

11.  How  many  eighths  in  |  and  |? 

12.  How  many  tenths  in  \  and  1? 

13.  How  many  twelfths  in  |  and  I? 
14    How  many  fifteenths  in  |  and  |? 

15.  How  many  sixteenths  in  |  and  g  ? 

16.  How  many  eighteenths  in  |  and  |  ? 

What  is  the  sum 

17.  Of  '  and  I?  21.  Off  and  §? 

18.  Of  l  and  I?  22.   Of  f  andf? 

19.  Of  4  and  if  23.  Of  |  and  |? 
20    O/  \  and  |?  24.  Off  and  «? 


56  MENTAL  ARITHMETIC. 

25  'Of  §  and  £?  36.  Of  2'  and  34  ? 

26  Of  |  and  |?  37.  Of  3j  and  4J  ? 

27.  Of  |  and  I?  38.  Of  2§  and  If? 

28.  Of  §  and  f  ?  39.  Of  34  and  2*  ? 

29.  Of  J,  i,  and  |?  40.  Of  6*  and  5§  ? 

30.  Of  i,  i,  and  £?  41.  Of  4J  and  5$? 

31.  Of  |,  |,  and  £  ?  42.  Of  6£  and  5|  ? 

32.  Of  |  and  |?  43.  Of  7-3  and  81? 

33.  Off  and  Ty  44.  Of  l|,  2f  ,  and  3|? 

34.  Of  |  and  |?  45.  Of  24,  3},  and  4bl  ? 

35.  Of  I  and  I?  46.  Of  3|,  2j,  and  4? 


|? 
llar; 


47.  Thomas  having  |  of  a  dollar,  found  f  of  a  dol 
how  much  had  he  then  ? 

48.  A  man  having  |  of  a  barrel  of  flour,  bought  |  of 
a  barrel  ;  how  much  had  he  then  ? 

49.  Mary  having  J  of  a  dozen  of  pins,  found  |  of  a 
dozen  ;  how  many  pins  had  she  then  ? 

50.  Peter  having  |  of  a  certain  sum  of  money,  found 
£  of  the  same  sum,  and  then  had  $21  ;  how  much  was 
the  sum  ? 

51.  |  of  a  certain  number,  increased  by  |  of  the  same 
number,  equals  34;  required  the  number. 

52.  Fanny's  number  of  roses,  increased  by  ^  and  -|  of 
her  number,  equals  55  ;  how  many  roses  has  she  ? 

53.  |  of  A's  money  increased  by  its  |,  equals  69  doll 
lars  j  how  much  money  has  A  ? 

54.  B  gave  $24  for  a  watch,  and  |  +  f  of  this  is  4 
times  what  he  paid  for  a  chain  ;  required  the  cost  of  the 
chain. 

55.  If  }  of  James's  age,  increased  by  ^  and  |  of  his 
age,  equals  57  years,  what  is  James's  age  ? 

Subtract 

56    l  from  ^.  61.  f  from  f 

57.  £  from  \.  62.  f  from  | 

58.  |  from  f  .  63.  |  from  f  . 

59.  i  from  \  64.  |  from  f  . 

60.  i  from  |  65.  4  from  I. 


MENTAL  ARITHMETIC.  57 

66  ^  from  |.  72.  §  from  f . 

67  |  from  |.  73.   |  from  |. 

.    68    |  from  f .  74.  2^  from  3f 

69    |  from  f.  75.  3£  from  4-]-. 

70.  |  from  f  76.  21  from  3£ 

71.  {  from  £.  77.  3£  from  5|, 

78  What  number  is  that,  which  being  diini Dished  by 
its  f ,  equals  36  ? 

79  Joshua's  age,  diminished  by  its  |,  equals  20  years  j 
how  old  is  Joshua  ? 

80.  A  certain  sum  of  money,  diminished  by  its  J  and  J, 
equals  15  dollars;  required  the  sum. 

81.  What  number  is  that,  which  being  increased  by 
its  1  and  diminished  by  its  |,  equals  130 ? 

82.  A  boy  having  36  marbles,  lost  |  of  them,  and  then 
found  |  as  many  as  he  had  at  first ;  how  many  had  he 
remaining? 

83.  $40  is  4  times  what  A  paid  for  a  chain,  and  the 
cost  of  the  chain,  increased  by  its  11  tenths,  is  {  of  the 
cost  of  his  watch;  required  the  cost  of  the  watch.. 

84.  Janson's  age,  diminished  by  its  \  and  1,  is  22 
years,  which  is  |  of  his  uncle's  age ;  required  the  age 
of  each. 

85.  |  of  B's  money,  increased  by  the  difference  be 
tween  |  and  J  of  his  money,  equals  190  dollars  \  required 
his  money. 

What  is  the  value 

86.  Of  1  +  4— |?  93.  Of  2-|  +  4^— 41? 

87.  Of  i  +  l—  i?  94.  Ofli  +  lJ—  2|? 

88.  Of  | +  |—i?  95.  Of3|  +  2|  — 51? 
89    On  +  |—  |?  96.  Of2i—  li  +  3|? 
90.  Of  l  +  l-i?  97.  Of  41—41  +  61? 
91    Of  ^  —  A  +  i?  98.  Of6i  — 21  +  31? 
92.  Of  |— &'+!?  99.  Of2|  +  2i—  2-Jj? 
100.  $60  is  |  of  what  B  gave  for  a  horse,  and  the  cost 

of  the  horse,  increased  by  its  3  fifths,  is  5  times  what  he 
paid  for  a  sleigh ;  required  the  cost  of  the  sleigh. 
6 


58  MENTAL  ARITHMETIC. 


LESSON  V. 

1.  A  man  spent  |  of  his  money  for  a  horse,  and  then 
had  $60  remaining ;  how  much  money  had  he  at  first  ? 

SOLUTION. — After  spending  1  of  his  money,  there  remained 
3  —  1  =  2  of  his  money,  which  equals  $60.  If  2  Of  his  money 
eqnals  $60,  |  equals  £.  of  $60,  which  is  $30,  &c. 

2.  A  lady,  after  giving  away  |  of  her  money,  had  only 
40  cents  remaining;  how  much  money  had  she  at  first  ? 

3.  A  farmer  sold  |  of  his  cows,  and  then  had  25  re- 
maining ;  how  many  had  he  at  first  ? 

4.  A  boy  lost  4  marbles  and  found  10,  and  then  had  | 
as  many  as  at  first ;  how  many  had  he  at  first ? 

5;  Mary  gave  ^  of  her  money  for  silk,  and  J  for  satin, 
and  had  10  dollars  remaining ;  how  much  had  she  at  first? 

6.  Philo  lost  $16  and  found  §6,  and  then  had  f  as 
much  as  he  had  at  first;  how  much  had  he  before  his 
loss? 

7.  |  of  the  length  of  a  pole  is  in  the  air,  \  in  the 
water,  and  10  feet  in  the  ground;  required  the  length  of 
the  pole. 

8.  Henry's  money,  diminished  by  its  J  and  J,  equals 
57  dollars ;  how  much  money  has  he  ? 

9.  Milo  lost  24  cents,  and  then  found  |  as  many  as  h-e 
lost,  and  then  had  only  |  of  what  he  had  at  first ;  how 
many  had  he  before  his  loss  ? 

10.  Mr.  A's  money,  increased  by  its  ^  and  £,  equals 
34  dollars  ;  how  much  money  has  he  ? 

11.  Says  B  to  C,  |  of  my  age  diminished  by  |  of  my 
age,  equals  12  years ;  how  old  was  he  ? 

12.  Peter  gave  10  cents  for  a  pie,  which  is  f  of  -J  of 
the  cost  of  his  supper;  required  the  cost  of  his  supper. 

13.  A   lady  being  asked  her   age,  replied   that  her 
daughter's  age  is  8  years,  which  is  |  of  ^  of  her  age; 
required  her  age. 


MENTAL  ARITHMETIC.  59 

11.  Frank,  after  spending  |  of  his  money,  found  that 
f!6  was  |  of  what  he  had  remaining;  how  much  money 
had  he  at  first  ? 

15.  |  of  an  army  was  killed,  |  taken   prisoners,  and 
800  escaped ;  of  how  many  men  did  the  army  consist  ? 

16.  When  E  was  married  he  was  27  years  old,  and  | 
of  his  age  was  4  years  more  than  |  of  his  wife's  age;  re- 
quired the  age  of  his  wife. 

17.  A  boy,  after  spending  \  of  his  money  for  candies, 
and  |  for  peaches,  found  that  20  cents  was  §  of  what  re- 
mained ;  how  much  money  had  he  ? 

18.  What  number  is  that,  which  being  increased  by 
its  ^,  and  that  sum  diminished  by  |  of  the  number,  the 
remainder  is  50  ? 

19.  Mr.  K's  hat  cost  $6,  which  was  $3  less  than  f  of 
the  cost  of  his  coat ;  required  the  cost  of  the  coat. 

20.  What  number  is  that  which  being  doubled,  and 
then  diminished  by  its  |,  equals  60  ? 

21.  |  of  30  cents  is  5  cents  less  than  |  of  what  a  bushel 
of  potatoes  cost ;  what  will  be  the  cost  of  5  bushels  at  the 
same  rate  ? 

22.  What  number  is  that,  which  being  tripled   and 
diminished  by  its  |,  the  remainder  will  equal  70  ? 

23.  A  merchant  sold  1  of  |  of  his  stock  in  a, month; 
how  many  thirds  of  j  of  his  stock  remained  ? 

24.  Alsop  having  |  of  an  apple,  gave  J  of  what  he  had 
to  Jane ;  what  part  of  %  of  an  apple  remained  ? 

25.  A  farmer  having  T9Q  of  a  ton  of  hay,  sold  |  of  it ; 
how  much  of  it  then  remained  ? 

26.  A  boy  having  ^  of  a  bushel  of  berries,  sold  |  of 
them ;  what  part  of  a  bushel  had  he  remaining  ? 

27.  A  fishing  rod  is  15  feet  long,  and  |  of  its  length 
lacks  3  feet  of  being  ^  of  the  length  of  tie  line ;  requijed 
the  length  of  the  line. 

28.  A  tree  is  60  feet  high,  which  is   |  of  ^  of  the 
length  of  its  shadow,  diminished  by  20  feet ;  required  the 
length  of  the  shadow. 

29.  10  dollars  is  |  of  what  M  paid  for  a  cow,  and  the 


60  MENTAL  ARITHMETIC. 

cost  of  the  eow  is  ^  of  the  cost  of  an  ox ;  required  the 
value  of  the  ox. 

30.   A  pen  cost  16  cents,  which  is  f  of  f  of  what  the 
pen  and  holder  both  cost ;  required  the  cost  of  the  holder 


LESSON  VL 

1    How  many  are  4  times  |  ? 
SOLUTION. — 4  times  Ji  are  12,  which  equals  3  or  1  * .     There- 

88  2  /^ 

fere,  &c. 

2.  How  many  are  3  times  |  ?  3  times  |  ? 

3.  How  many  are  4  times  |  ?  5  times  -fa  ? 

4.  How  many  are  7  times  r\  ?  3  times  |  ? 

5.  How  many  are  4  times  f6^  ?  6  times  T72  ? 

How  many  are 

6.  3  times  f  ?  14.  4  times  f? 

7.  8  times  f  ?  15.  8  times  |? 

8.  5  times  |?  16.  5  times  |? 

9.  3  times  |?  17.  4  times  2^? 
10.  5  times  |?  18.  6  times  3^? 
11    6  times  f  ?  19.  8  times  5|? 

12.  10  times  &  20.  9  times  2f  ? 

13.  8  times  T\?  21.  10  times  4|? 

22.  If  you  give  to  8  ooys,  each  2|  apples,  how  many 
apples  will  it  require  ? 

23.  How  many  dollars  does  that  man  give  away,  who 
gives  to  each  of  10  beggars  |  of  a  dollar  ? 

24.  Mary  gave  to  each  of  12  poor  children,  |  of  a  pity 
and  had  3  pies  remaining;  how  many  pies  had  she? 

25.  How  many  quarts  of  nuts  will  it  require,  to  give 
|  of  a  quart  to  each  of  18  boys,  and  have  2  quarts  re- 
maining ? 

26.  What  will  be  the  cost  of  25  pounds  of  sugar,  at 
nhe  rate  of  |  of  a  dime  a  pound? 


MENTAL  ARITHMETIC, 


61 


27.  How  much  will  4|  yards  of  muslin  cost,  at  the 
rate  of  6  cents  a  yard  ? 

28.  What  will  15  books  cost,  at  the  rate  of  |  of  a  dol- 
lar apiece  ? 

29.  How  much  will  9  inkstands  cost,  at  the  rate  of  2 
for  |  of  a  dollar  ? 

30.  If  4  apples  cost  |  of  a  dime  what  will  16  applee 
cost  ? 

31 .  What  will  7  quarts  of  beans  cost,  if  4  quarts  cost 
^  of  a  dollar? 

32.  If  a  man  walk  3^  miles  in  2  hours,  how  far  wili 
ho  walk  in  1  day  of  10  hours  ? 

33.  William  lost  2J  dollars,  and  then  having  found  4 
times  as  much  as  he  lost,  had  12  dollars ;  how  much  had 
he  at  first  ? 

34.  Mary  found  4|  shillings,  and  then  losing  J  as  much 
as  she  found,  had  6-J  shillings  remaining ;  how  many  had 
she  at  first  ? 

35.  Since  2  times  |  ==  |,  or  |,  how  may  the  last  result 
be  obtained  by  omitting  the  analysis  ? 

36.  Since  3  times  |  =  lg,  or  |,  how  may  the  last  result 
be  obtained  without  the  analysis  ? 

37.  How  then  may  a  fraction  be  multiplied  by  a  num- 
ber which  will  divide  the  denominator  ? 

38.  What  is  the  effect  of  dividing  the  denominator  of 
a  fraction  by  any  number  ? 


How  many  are 

39. 

4 

times 

f 

49. 

10  times  ^ 

40. 

6 

times 

I? 

50, 

8  times  ||? 

41 

7 

times 

51. 

6  times  ||? 

42 

5 

times 

_9_? 

52. 

4  times  ||? 

43 

8 

times 

jor 

53 

3  times  2|? 

44. 

8 

times 

3? 

54 

5  times  2|? 

45. 

9 

times 

£{>? 

55. 

7  times  Si? 

^6. 

7 

times 

18? 

56. 

12  times  5\ 

47. 

6 

times 

-§!  f 

57. 

10  times  8^ 

48. 

4 

f  i 

times 

& 

58 

11  times  2^ 

62.  MENTAL  ARITHMETIC, 

59.  3  times  2f  miles  is  twice  the  distance  from  the 
Normal  School  to  Lancaster;  required  the  distance. 

60.  5  times  4|  miles  is  3  times  the  distance  from  Paoli 
to  Oakland;  required  the  distance. 

^61    A  is  6|  years  old,  and  8  times  A's  age  is  5  times 
B's  age ;  required  the  age  of  B. 

62.  Six  times  2|  miles  is  7?0  of  the  distance  from  Oak- 
land to  Christiana ;  what  is  the  distance  ? 

63.  6|  dollars  is  ^  of  4  times  3|  times  B's  money;  how 
much  money  has  B? 

64.  1  times  3j  miles  is  5^  times  the  distance  from 
Columbia  to  Marietta;  what  is  the  distance? 

65.  James  lost  $5|,  and  then  had  5  times  $6/3  re- 
maining; how  much  money  had  he  at  first? 

66.  Peter  gave  £  of  his  marbles  to  Samuel,  and  2  times 
J  of  them  to  Anson,  and  had  15  remaining;  how  many 
had  he  at  first  ? 

67.  A  having  lost  ^  of  his  hens,  sold  3  times  |  of  the 
remainder,  and  then  had  20 ;  how  many  had  he  at  first  ? 

68.  4 £  miles,  the  distance  from  M  to  B,  being  multi- 
plied by  8,  lacks  2  miles  of  being  2  times  the  distance 
from  M  to  W;  required  tH  distance. 


LESSON  VII. 

1.  What  is  |  of  4? 

SOLUTION.  —  J  of  1  is  £,  and  £  of  4  is  4  times  £,  which  are  j 
Therefore  ^  of  4  is  ^  of  one. 

2    What  is  |  of  5?  {of  6? 

3.  Whatip  lof  7?  J  of  9? 

4.  What  is  £  of  5?  \  of  10? 

5.  What  is  4  of  12?  J  of  20? 

6.  Whath       of  24?  J3of  32? 


A 
| 


7.  What  is  |  of  6?  $  of  10* 


MENTAL  ARITHMETIC.  68 

8.  What  is  4  of  3?  f  of  14? 

9.  What  is  |  of  4?  |  of  15? 
10    What  is  f  of  2?  f  of  12? 

11.  What  is  |  of  9?  |  of  16? 

12.  What  is  |  of  6?  |of20? 

13.  What  is  -^  of  5  ?  T»2  of  18  ? 

14    B  has  5  dollars,  and  |  of  B's  money  equals  ^  ct 
A's;  how  much  money  has  A? 

15.  James  is  7  years  old,  and  |  of  his  age  equals  J  of 
Mary's  age;  how  old  is  Mary? 

16.  One  part  of  a  pole  is  9  feet  long,  which  lacks  8 
feet  of  being  T4T  of  the  length  of  the  other  part ;  required 
the  length  of  the  pole. 

17.  The  distance  from  A  to  B  is  11  miles,  and  f  of 
this  is  \^  of  the  distance  from  A  to  C ;  required  the  dis- 
tance to  C. 

18.  A  watch  cost  $21,  and  4  of  its  cost  is  T7^  of  the 
cost  of  the  chain;  required  the  cost  of  both. 

19.  A  cask  contains  5  gallons,  and  fy  of  its  contents  is 
\  |  of  the  contents  of  another  cask ;  required  the  contents 
of  the  second  cask 

20.  Charles  has  9  apples,  and  |  of  his,  -|-  |  of  an  apple, 
equals  f  of  Chester's;  how  many  has  Chester? 

21.  What  is  |  off? 

SOLUTION. — 1  of  £  is  2,  and  if  4  of  &  is  2,  5  of  &  are  2  times 

388  38838 

§,  •which  are  i,  or  1      Therefore,  &e. 

b  82 

What  is 
22    |  of  48?  -30    |  of  2?? 

23.  fof|?  31    Irfilf 

24.  jof  j?  32.  i  of  Sir 

25.  |ofA§?  33.  |ofl|? 

26.  I  of  li?  34.  2J  times  1|? 

27.  f  ofj|?  35.  24  times  Si? 

28.  |  of -j%?  36.  3|  times  3 A? 

29.  f  of2|?_  37.  2|  times  2f? 

38.  j  of  $3|  is   £  of  what  A  gave  for  a  bureau ;  whal 
was  the  cost  of  the  bureau  ? 


64  MENTAL  ARITHMETIC. 

39.  |  of  $ii  is  ^j  of  the  cost  of  a  watch;  now  much 
did  the  watch  costf 

40.  B  has  27  marbles,  and  f  of    -g6  of  B's  number 
equals  |  of  C's ;  how  many  has  C  ? 

41.  Henry's  hat  cost  i  of  an  eagle,  which  is  f  of  f  of 
the  cost  of  his  coat ;  required  the  cost  of  the  coat. 

42.  Peter  had  |  of  a  barrel  of  flour,  and  after  selling 
|  of  it,  the  remainder  is  Taff  of  what  Paxton  had ;  how 
much  had  Paxton  ? 

43.  If  there  are  48  chestnuts  in  a  pint,  how  many 
does  each  of  two  boys  receive,  if  A  receives  |  of  a  pint, 
and  B  |  as  many  as  A  ? 

44.  Mary  bought  |  of  a  paper  of  needles,  which  is   | 
of  what  Sarah  bought;  how  many  did  each  purchase,  pro- 
vided there  are  24  needles  in  a  paper? 

45.  Frank  has  18   apples,   Francis   |  as  many,  and 
Fanny  |  as  many  as  Francis ;  how  many  has  each  ? 

46.  A  has  40  fruit  trees,  T40  of  which  bear  apples,  ^ 
of  the  remainder,  pears,  and  the  rest,  peaches;  how  many 
trees  of  each  kind  has  he  ? 


LESSON  VIII. 

1.  What  is  |  off? 

SOLUTION. — J  of  £  is  one  of  the  three  equal  parts  into  which 
J  may  be  divided  ;  if  each  fourth  is  divided  into  three  equal 
parts,  4-  fourths  or  the  unit  will  be  divided  into  4  times  3,  or  12 
equal  parts ;  hence,  each  part  is  J2-  of  a  unit.  Therefore  J  of 

i  1S  i1*- 

2.  What  is  4  of}?  }  of}?  Jof  }? 
3    What  is  I  of  i  ?  i  of  j  ?  j  of  J  ? 

4.  What  is  }  of  i  ?  I  of  |  ?  J  of  }  ? 

5.  What  is  l  of  ^?  i  of  I?  -l  of  $? 
f>.  What  is  }  of  l  ?  I  of  1  ?  %  of  £  ? 

7  What  is  i  of  TV  ?  £  of  *  ?  i  of  ^  ? 

8  What  is  4  of  TV  ?  ^  of  TV  ?  JT  of  &  ? 


MENTAL  ARITHMETIC.  65 

9.  What  is  J  of  4  ?  j  of  J-j  ?  i  of  -^  ? 

10.  Mary  having  ^  of  a  pie,  gave  |  of  it  *o  Hannah  j 
what  part  of  a  pie  did  Hannah  receive  ? 

11.  Philo  having  i  of  an  orange,  gave  |  of  it  to  Peter; 
what  part  of  an  orange  did  Peter  receive : 

12.  A  had  |  of  a  dollar,  and  gave  £  of  it  to  B ;  what 
part  of  a  lo/lar  did  B  receive  ? 

13.  Since  |  of  \  equals  yV,  how  may  the  same  result 
be  obtained  without  the  analysis  ? 

14.  Since  \  of  £  equals  ^Q,  how  may  we  obtain  the 
same  result  without  the  analysis  ? 

15.  What  is  4  of  i  ?  |  of  J  ?  J  of  *  ?  I  of  4  ? 

16.  What  is  |  of  J  ?  i  of  £  ?  4  of  -j  ?  £  of  $  ? 

17.  What  is  *  of  i?  4  of  4?  $of  i?  TV°fTV? 

NOTE. — Problem  1st  may  also  be  solved  thus:  J  equals  ^, 
and  J  of  fy  is  y1^,  hence  J  of  J  is  T^.  This  is  simpler  than 
the  solution  given,  but  since  it  does  not  show  the  reason  why 
J  of  £  =  y1^,  we  prefer  the  other  solution. 

18.  What  is  the  difference  between  A  -of  -*,  and  |  of  {  ? 

19.  What  is  the  value  of  |  of  ^,  subtracted  from  5  of  |  ? 

20.  How  much  greater  is  |  of  ^,  than  ^  of  £  ? 

21.  A  man  owning  £  of  a  farm,  sold  ^  of  it  to  his 
aeighbour ;  what  part  of  it  did  his  neighbour  receive  ? 

22.  Susan  bought  £  of  a  cake,  and  gave  Eliza  |  of  it; 
how  much  did  Susan  retain  ? 

.23.  A  man,  having  4  of  a  share  of  bank  stock,  sold  | 
of  it  j  how  much  of  a  share  did  he  retain  ? 

24.  Carlo,  finding  j  of  a  pound  of  meat,  let  Towser  eat 
i  of  it ;  what  part  of  a  pound  did  Carlo  eat  ? 

25.  'What  is  A's  age,  supposing  that  -J  of  10  years  ;«  J 
of  -i  of  his  age  ? 

26.  How  much  money  has  Jacob,  if  \  of  13  dollars  is 
|  of  |  of  his  money  ? 

27.  A  cow  cost  17  dollars,  and  |  of  the  cost  of  the  cow 
is  |  of  -|  of  the  cost  of  an  ox ;  required  the  cost  of  the  ox 

28.  A  pe«i  cost  20  cents,  and  f  of  its  cost  is  |  of  f  of 
the  cost  of  the  holder;  required* the  cost  of  the  holder. 


MENTAL  ARITHMETIC. 

29.  What  is  J  of  fj  ? 

SOLUTION.— i  of  1  is    1  ,  and  if  1  of  i  is  _^,  1  of  4  is  4  times 

3          5          15  3  o          1  o '    3          o 

i     which  are  _4^.,     Therefore  JL  of  4  —   4 . 

}  Sr  J5  3515 

30.  What  is  |  of  |  ?  \  of  |  ?  \  of  f  ? 

31.  What  is  \  of  f  ?  J  of  f  ?  |  of  |  ? 

32.  What  is  I  of  f  ?  i  of  |  ?  I  of  -^  ? 
33  What  is  |  of  |  ?  f  of  f  ?  4  Of  f  ? 
34,  What  is  I  of  |  ?  |  of  f  ?  f  of  f  f 

35  What  is  f  of  -f  ?  T7^  of  |  ?  f-  of°T7^  ? 

36  ^yhat  is  T80  of  ft  ?  |  of  T65  ?  f  of  fg  ? 

37.  Since  f  of  |  =  -,8^,  in  what  manner  may  we  obtain 
the  same  result  by  omitting  the  analysis  ? 

38.  What  is  |  of  §  ?  f  of  f  ?  |  of  -?-  ?  i  of  f  ? 

39.  What  is  f  of  |  ?  |  of  |  ?  |  of  |  ?  f  of  T%  ? 

40.  WThat  is  |  of  |  ?  4  of  L0  ?  f  of  ^  ?  7  of  -£ 

41.  What  is  f  of  y  ?°|  Of  e  ?  |  of  j|  ?  ff  of  _ 

42.  Johnston,  having  |  of  a  melon,  gave  |  of  it  to 
Martin ;  how  much  of  it  remained  ? 

43.  Having  |  of  a  bushel  of  apples,  I  gave  |  of  them 
to  Westlake ;  what  part  of  a  bushel  remained  ? 

44.  Wrhat  is  the  difference  between  |  of  |  of  a  lemon, 
and  |  of  |  of  a  lemon  ? 

45.  How  much  money  has  Sarah,  if  |  of  11  dollars  is 
|  of  her  money  ? 

46.  A  can  build  a  boat  in  2|  weeks,  which  is  |  of  | 
of  the  time  in  which  B  can  build  it;  how  long  will'  it  take 
B  to  build  it? 

47.  A  hat  cost  $5,  and  f  of  its  cost  is  f  of  JJ  of  the 
cost  of  a  coat ;  required  the  cost  of  the  coat. 

48  A  man  lost  |  of  his  money,  and.  then  found  as 
much  as  J  of  the  remainder;  what  part  of  his  money  did 
he  then  have  ? 

49.  A  squirrel  fell  |  of  the  distance  from  the  top  of  a 
tree  tc  the  ground,  and  then  ascended  \  of  the  distance 
he  was  from  the  ground ;  what  part  of  the  whole  dis- 
tance was  he  from  the  ground? 

50.  Annie,  having  f  of  a  pound  of  candies,  shared 


MENTAL  ARITHMETIC.  67 


t.hein  equally  with  5  of  her  schoolmates)  what  part  of  a 
pound  did  each  receive? 

51.  Richard,  having  J  of  f  of  a  quart  of  chestnuts, 
shared  them  equally  with  4  of  his  playfellows ;  what  part 
of  a  quart  did  each  receive  ? 

52.  Matthew  lost  |  of  his  property,  then  selling  \  of 
the  remainder,  bought  as  much  as  1  third  of  what  then 
remained ;  what  part  of  his  property  did  he  then  have  ? 

53.  A,  having  f  of  a  dollar,  gave  i  of  it  to  B,  and  B 
gave  ^  of  his  to  0 ;  what  part  of  a  dollar  did  each  then 
have  ? 

54.  A  kite  in  the  air  fell  ^  of  the  distance  to  the  ground, 
tli en  arose  |  of  the  distance  it  was  from  the  ground,  and 
then  fell  \  of  the  distance  it  arose;  what  part  of  the 
whole  distance  was  it  from  the  ground  ? 


LESSON  IX. 

1.  How  many  times  is  f  contained  in  4  ? 

SOLUTION. — 1  is  contained  in  4,  4  times;  and  if  1  is  contained 
in  4,  4  times,  J  is  contained  in  4,  3  times  4  times,  which  are  12 
times,  and  2  thirds  is  contained  in  4  J  of  12  times,  or  6  times, 
Therefore.  &c. 

2.  How  many  times  is  |  contained  in  2  ?  In  3  ? 

3.  How  many  times  is  |  contained  in  3  ?  In  5  ? 

4.  How  many  times  is  |  contained  in  2  ?  In  4  ? 

5.  How  many  times  is  |  contained  in  5  ?  In  7  ? 

6.  How  many  times  is  |  contained  in  4  ?  In  5  ? 

7.  How  many  times  is  |  contained  in  2  ?  In  4  ? 

8.  How  many  times  is  |  contained  in  5  ?  In  8  ? 

9.  How  many  times  is  -$$  contained  in  7  ?     In  9  ? 

10.  How  many  times  is  T8T  contained  in  2  ?     In  4  ? 

11.  How  many  times  is  |  contained  in  4  ?     In  12  ? 

12.  By  what  method  may  we  derive  the  result^  obtained 
above,  without  the  analvsis? 


68  MENTAL  ARITHMETIC. 

13    If  a  yard  of  cloth  cost  |  of  a  dollar,  how  manj 
yards  can  you  buy  for  12  dollars  / 

14.  If  2  quarts  of  nuts  cost  j  of  a  dime,  how  many 
quarts  can  be  bought  for  1  dollar: 

15.  How  many  yards  of  tape  can  be  bought  for  11 
cents,  if  3  yards  cost  2^  cents  ? 

16.  What  cost  7  peaches,  at  the  rate  of  5  peaches  for 
6 1  cents? 

17.  If  5  pints  of  milk  cost  12  cents,  how  many  pints 
can  you  purchase  for  25  cents  ? 

18.  If  2^  barrels  of  apples  cost  15  dollars,  how  many 
barrels  can  be  bought  for  8|  dollars  ? 

19.  If  |  of  an  apple  cost  j  of  a  cent,  what  will  £  of  | 
of  an  appk  cost  ? 

20.  What  cost  2£  oranges,  at  the  rate  of  4  oranges  for 
51  cents? 

21.  If  3|  boxes  of  butter  cost  $5$,  what   will   10 
boxes  cost,  at  the  same  rate  ? 

22.  7^  dollars  is  ^  of  5  times  what  a  watch-key  cost, 
and  the  chain  cost  8  times  as  much ;  required  the  cost  of 
the  chain. 

23.  What  cost  16  lemons,  if  5-J  lemons  cost  12|  cents? 

24.  What  cost  17  yards  of  lace,  if  4|  yards  cost  7^ 
dimes  ? 

25.  If  3  men  can  do  a  piece  of  work  in  6f  days,  how 
long  will  it  take  12  men  to  do  the  same? 

26.  If  it  require  9  men  to  do  a  piece  of  work  in  4| 
days,  how  many  men  will  be  required  to  do  it  in  13 
days  ? 

27.  What  cost  26  gallons  of  molasses,  if  4|  gallois 
cost  2^  dollars? 


28.  How  many  times  is  f  contained  in  |? 

REMARK. — Solution  similar  to  that  at  the  beginning 
>son. 

29.  How  many  times  is  |  contained  in  J  ?     In 


MENTAL  ARITHMETIC.  69 

30  How  many  times  is  |  contained  in  f?  In  |? 

81.  How  many  times  is  |  contained  in  |?  In  |? 

32.  How  many  times  is  |  contained  in  |  ?  In  |  ? 

33.  How  many  times  is  |  contained  in  f?  In  |? 

34.  How  many  times  is  |  contained  in  |  ?  In  -3$  ? 

35.  How  many  times  is  |  contained  in  |?  In  |? 

36.  How  many  times  is  g  contained  in  i§?     In  ||F 

37.  How  many  times  is  ^  contained  in  28T?     ID  i|? 

38.  How  many  times  is  |  contained  in  T7g?     In  ||? 

39.  How  many  times  is  |  contained  in  J'/? 

ANOTHER  SOLUTION. — 3  is  equal  to  ^fl,  and  1  is  equal  to  4  £  • 
10  is  contained  as  many  times  in  1 1  as  10  is  contained  in  12, 
grhich  is  1*2  or  6  times.  Therefore,  &c. 

1  0          5 

40.  How  many  times  is  f  contained  in  |  ?     In  |  ? 

41.  How  many  times  is  f  contained  in  |?     In  ^? 

42.  How  many  times  is  |  contained  in  |?     In  |? 

43.  How  many  times  is  g  contained  in  |?     In  |? 

44.  How  many  times  |  is  4  ?     |  ?     |  ? 

45.  How  many  times  |  is  |?     f?     T%? 

46.  How  many  times  |is|?     |?     J? 

47.  How  many  times  |  is  \ •  ?     |  ?     |  ? 

48.  How  many  times  |  is  |  ?     |?     T^j? 

49.  From  the  results  in  either  of  the  above  solutions, 
what  method  may  be  derived  to  divide  fractions  by  omit- 
ting the  analysis  ? 

50.  If  a  yard  of  muslin  cost  |  of  a  dime,  how  many 
yards  can  you  purchase  for  |  of  a  dime  ? 

51.  If  a  quart  of  vinegar  cost  T30  of  a  dollar,  how  many 
quarts  can  you  buy  for  2^  dollars  ? 

52.  How  many  pens,  at  ^  of  a  dime  each,  can  be  bought 
for  10  sheets  of  paper,  worth  |  of  a  cent  each  ? 

53.  A  divided  14  apples  equally  among  his  compan- 
ions, giving  to  each  |  of  an  apple ;  required  the  number 
of  companions. 

54.  A  lady  distributed  29  dimes  equally  among  some 
poor  children,  giving  to  each  5|  dimes;  hew  many  chil- 
dren were  there  ? 


70  MENTAL  ARITHMETIC. 

55.  How  many  bushels  of  grain,  at  $|  a  bushel,  can  be 
bought  for  12  bushels  of  apples,  worth  $|  a  bushel  ? 

56.  If  1  pound  of  butter  cost  $|,  how  many  pounds 
can  be  had  for  |  of  a  barrel  of  flour,  worth  $6  per  barrel  ? 

57.  B  bought  6  yards  of  ribbon,  worth   5|   cents   a 
yard;   how  many  apples,  worth  1|   cents  each,  will  be 
required  to  pay  for  it  ? 

58.  Harvey  bought  8  bushels  of  potatoes,  worth  $|  a 
bushel,  and  paid  for  them  with  eggs,  wortli  $|  a  dozen ; 
how  many  eggs  did  it  take  ? 


LESSON  X. 

1.  What  part  of  2  is  £? 

SOLUTION.— 1  is  £  of  2,  and  £  is  J  of  |  of  2,  which  is  J  of  2, 
and  |  is  3  times  £,  or  f  of  2.     Therefore  f  is  f  of  2. 

2.  What  part  of  3  is  f  ?     Of  2  is  f  ? 

3.  What  part  of  4  is  §  ?     Of  5  is  5  ? 

4.  What  part  of  4  is  |  ?     Of  7  is  g  ? 

5.  What  part  of  9  is  f  ?     Of  5  is  I  of  |? 

6.  What  part  of  6  is  f  of  f  ?     Of  7  is  f  of  I  ? 

7.  What  part  of  2  is  f  of  f  ?     Of  5  is  |  of  fg  ? 

8.  What  part  of  6  is  |  of  §?     Of  8  is  f  of  j ? 

9.  If  a  pole  10  feet  long  cast  a  shadow  3|  feet,  what 
is  the  length  of  the  shadow  of  a  pole  8  feet  long,  at  the 
earne  time  of  day? 

10.  If  7  apples  are  worth  5|  peaches,  how  many  apples 
are  12  peaches  worth  ? 

11.  If  at  a  certain  time  of  day,  a  pole  9  feet  long  cast 
a  shadow  4|  feet,  what  must  be  the  length  of  a  pole  to 
cast  a  shadow  54  feet  long,  at  the  same  time  of  day  ? 

12.  If  8  pipes  fill  a  vessel  in  2£  hours,  how  many  pipes 
will  be  required  to  fill  it  in  y1^  of  an  hour  ? 

13.  Required  the  length  of  the  shadow  of  a  pole,  16 


MENTAL  ARITHMETIC.  71 

feet  long,  at  the  same  time  that  a  pole  3|  feet  long  casts 
a  shadow  7|  feet  in  ength  ? 

14.  Mr.  B,  having  lost  $10,  found  that  only  g  of  hia 
money  remained;  how  much  money  had  he? 

15.  A  merchant  sold  goods  for  I|  of  what  they  cost, 
and  thereby  lost  $24;  what  was  the  cost  of  his  goods? 

16.  What  part  of  f  is  f  ? 

SOLUTION. — J  is  J  of  f,  and  |  or  one  is  3  times  j,  or  f  of  f . 
Since  one  is  f  of  f ,  £  is  \  of  f ,  which  is  j3^  of  f ,  and  f  is  4  times 
f0t  which  are  |J  or  f  of  f.  Therefore,  f  is  f  of  f. 

What  part 

17.  Of  |  is  §?  24.  Of  |  is  I? 

18.  Of  |  is  |  ?  25.  Of  |  is  |  ? 

19.  Of  I  is  I?  26.  Of  f  is  2'? 

20.  Of  f  is  |  ?  27.  Of  f  is  |  of  §  ? 

21.  Of  f  is  I?  28.  Of  J  is  I  of  I? 

22.  Of  1  is  |  ?  29.  Of  -fa  is  |  of  I  ? 

23.  Of  f  is  |?  30.  Of  \\  is  |  of  2|? 

31.  16  is  |  of  how  many  times  5? 

32.  18  is  |  of  how  many  times  8? 

33.  25  is  |  of  how  many  times  10? 

34.  14  is  y7^  of  how  many  times  4? 

35.  27  is  |  of  how  many  times  9  ? 

36.  15  is  |  of  how  many  times  3? 

37.  28  is  |  of  how  many  times  7  ? 

38.  10  is  T52  of  how  many  times  12? 

39.  20  is  |  of  how  many  times  \  of  10? 

40.  24  is  |  of  how  many  times  f  of  12? 
28  is  ^  of  how  many  times  \  of  21  ? 
30  is  |  of  how  many  times  J  of  12  ? 
18  is  |  of  how  many  times  ^  of  14  ? 
40  is  |  of  how  many  times  |  of  30? 
36  is  |  of  how  many  times  |  of  15? 
60  is  |  of  how  many  times  |  of  14  ? 

4-8   ia    8    r\f   hrinfr  momr  •Hmoa    3    r\f    I  r\r 


40. 
41. 
42. 
43. 
44. 
45. 
46. 
47. 
48. 

49.  60  is  |  of  how  many  times  ^  ui  ^v : 

50.  80  is  |  of  how  many  times  §  of  30? 


48  is  |  of  how  many  times  |  of  16? 
35  is  |  of  how  many  times  -|  of  18? 
60  is  -§  of  how  many  times  |  of  20  ? 


72  MENTAL  ARITHMETIC. 

51.  T|  of  16  is  how  many  times  ^  of  12? 

52.  |  of  30  is  how  many  times  |  of  10? 

53.  |  of  40  is  how  many  times  f  of  21  ? 

54.  |  of  45  is  how  many  times  |  of  15? 
*  of  42  is  how  many  times  ^  of  14  ? 

of  48  is  how  many  times  -|  of  16? 
|  of  56  is  how  many  times  T6T  of  22  ? 
|  of  80  is  how  many  times  |  of  25  ? 

59.  |  of  72  is  how  many  times  |  of  16? 

60.  |  of  27  is  how  many  times  f  of  12? 

61.  A's  horse  cost  $200,  and  |  of  this  is  twice  the  cost 
of  his  sleigh,  and  the  sleigh  cost  4  times  as  much  as  his 
harness ;  required  the  cost  of  each. 

62.  B's  wedding-coat  cost  $40,  and  |  of  this  is  twice 
the  cost  of  his  vest,  and  three  times  the  cost  of  his  hat ; 
what  was  the  cost  of  each  and  of  all  ? 

63.  Hannah's  wedding-dress  cost  $50,  and  J  of  this  is 
4  times  the  cost  of  her  bonnet,  and  |  of  the  cost  of  her 
cloak;    required  the  cost  of  the  bonnet  and  cloak,  re- 
spectively. 


LESSON  XI, 

1.  Myron,  having  f  of  a  certain  sum  of  money,  found 
|  of  the  same  sum,  and  then  had  $28 ;  required  the  sum. 

2.  If  |  of  John's  age,  increased  by  \  and  \  of  his  age, 
equals  34  years,  what  is  his  age? 

3.  What  number  is  that,  which  being  increased  by  its 
£,  and  diminished  by  its  |,  equals  35? 

4.  Henry's  age,  diminished  by  its  \  and  \,  equals  15 
years,  and  his  age  is  f  of  his  brother's  age;  required  the 
age  of  each. 

5.  Francis,  after  losing  f  of  his  money,  found  that  $1'J 
was  |  of  what  remained ;  how  much  money  had  he  ? 

6.  What  is  the  sum  of  the  fractions  £,  |,  £,  and  \  ? 


MENTAL  ARITHMETIC.  73 

7.  How  many  apples  does  that  man  give  away,  who 
gives  to  5  girls  each  |  of  an  apple? 

8.  What  is  the  cost  of  7  pens,  at  the  rate  of  |  of  a 
cent  apiece  ? 

9.  B  is  9|  years  old,  and  4  times  B's  age  is  |  of  C's 
ag'*;  what  is  the  age  of  C? 

10.  Peter  gave  ^  of  his  money  to  James,  and  3  times 
|  of  it  to  John,  and  then  had  $20  remaining;  how  much 
money  had  Peter  at  first? 

11.  A  chain  cost  $15,  and  £  of  its  cost  is  |  of  the  cost 
of  a  watch  ;  required  the  cost  of  the  watch. 

12.  Required  the  value  of  |  of  |,  |  of  i,  i  of  i,  and 

t°ft? 

13.  A  cow  cost  $14,  and  -|  of  the  cost  of  the  cow  is  | 
of  the  cost  of  a  horse ;  required  the  cost  of  the  horse. 

14.  A,  having  $£,  gave  ^  of  it  to  B,  and  B  gave  \  of 
his  part  to  C ;  what  part  did  each  then  have  ? 

15.  If  a  yard  of  cloth  cost  |  of  a  dollar,  how  many 
yards  can  be  bought  for  9  dollars  ? 

16.  How  many  yards  of  muslin  can  be  bought  for  $6, 
if  2  yards  cost  |  of  a  dollar  ? 

17.  How  many  times  |  is  |?    How  many  times  ^  is  ^? 

18.  If  ^  of  an  apple  cost  |  of  a  cent,  what  will  i  of  an 
apple  cost  ? 

19.  What  cost  30  yards  of  lace,  if  3|  yards  cost  |  of  a 
dollar  ? 

20.  If  3  men  can  do  a  piece  of  work  in  3|  days,  how 
long  will  it  take  5  men  to  do  it? 

21.  If  7  men  can  do  a  piece  of  work  in  2^  days,  how 
long  will  it  require  6  men  to  do  it? 

22.  If  a  yard  of  muslin  cost  f  of  a  dime,  how  much 
can  you  buy  for  |  of  a  dime? 

23.  What  part  of  f  is  §  ?     What  part  of  1£  is  2-J  ? 

24.  A  shared  8  apples  with  his  companions,  giving  to 
each  |  of  an  apple ;  required  the  number  of  companions. 

25.  How  many  bushels  of  corn,  worth  5|  shillings  a 
bushel,  can  be  bought  for  34  shillings? 

7* 


74  MENTAL  ARITHMETIC. 

26    How  many  apples  will  pay  for  10  peaches,  if  5 
apples  are  worth  8-i  peaches  ? 

27.  Mary   shared   21    dimes   with    her   schoolmates, 
giving  to   each  2±  dimes;   how  many  schoolmates  had 
she? 

28.  B  bought  4  yards  of  silk,  worth  $1|  a  yard,  and 
paid  for  it  with  cloth,  worth  $11  a  yard;  how  many  yards 
of  cloth  did  it  take  ? 

29.  The  distance  from  Paoli  to  Christiana  is  24  mites, 
and  |  of  this  distance  is  |  of  the  distance  from  Christiana 
to  Lancaster;  what  is  the  distance  to  Lancaster? 

30.  The  distance   from  Columbia  to  Rockville  is  30 
miles,  and  |  of  this  distance  is  |  of  the  distance  from 
Columbia  to  Newport;  required  the  distance  to  Newport. 

31.  The  distance  from  Conewago  to  Duncannon  is  3 
miles,  and  |  of  this  distance  is  ^  of  the  distance  from 
Conewago  to  Mexico ;  what  is  the  distance  to  Mexico  ? 


LESSON  XII. 

1.  Cyrus,  after  spending  |  of  his  fortune,  found  that 
$40  was  |  of  what  remained ;  wh£t  was  his  fortune  ? 

2.  What 'is  the  length  of  a  pole,  the  shadow  of  which 
is  10  feet,  at  the  same  time  that  a  pole  10  feet  long  casts 
a  shadow  2^  feet  in  length? 

3.  Martha  is  35  years  old,  and  |  of  her  age  is  £  of 
twice  her  son's  age;  how  old  is  her  son  ? 

4.  Rachael  had  |  of  a  peck  of  walnuts,  which  she 
shared  with  5  of  her  schoolmates ;  what  part  of  a  peck 
did  each  receive? 

5.  Henry,  having  |  of  a  barrel  of  cider,  sold  his  neigh* 
bour  J  of  it ;   what  part  of  a  barrel  remained  ? 

6.  A  farmer,  having  sold  ^  of  his  Sheep,  and  10  cows, 
found  15  sheep  and  -J  of  his  cows  remained;  how  many 
of  eaah  di  1  he  own? 


MENTAL  ARITHMETIC.  75 

7.  Take  any  number,  multiply  it  by  4,  divide  by  2, 
multiply  by  6,  divide  by  3,  then  divide  by  twice  the 
number,  and  name  the  result. 

8.  A's  money,  being  increased  by  its    J,  and  theD 
diminished  by  its  ^,  is  $40;  required  his  money. 

9.  |  of  B's  money,  increased  by  tke  difference  between 
|  and  |  of  his  money,  is  $55 ;  what  is  his  money  ? 

10.  When  B  was  married  he  was  25  years  old,  and  | 
of  his  age  was  3  years  more  than  |  of  his  wife's  age; 
required  the  age  of  his  wife. 

11.  What  number  is  that,  which  being  diminished  by 
its  |,  and  the  remainder  increased  by  its  |,  equals  40? 

12.  Stephen  lost  12  cents,  then  found  ^  as  much  as  he 
lost,  and  then  had  j  as  much  as  he  had  at  first ;  how  much 
had  he  at  first? 

13.  A  fishing-rod  is  16  feet  long,  and  |  of  its  length 
lacks  2  feet  of  being  |  of  the  length  of  the  line ;  -required 
the  length  of  the  line. 

14.  A  watch  cost  $40,  which  is  |  of  |  of  the  cost  of 
the  watch  and  chain ;  required  the  cost  of  the  chain. 

15.  Susan  has  7  peaches,  and  |  of.  Susan's  number, 
minus  |  of  a  peach,  is  |  of  Elizabeth's  number;  how 
many  has  Elizabeth  ? 

16.  If  there  are  50  chestnuts  in  a  pint,  how  many  do 
A  and  B  receive  respectively,  if  A  has  |  of  a  pint,  and  B 
|  as  many  as  A  ? 

17.  A  has  60  fruit  trees,  |  of  which  bear  peaches,  § 
of  the  remainder,  pears,  and  the  remainder,  apples ;  how 
many  are  there  of  each  ? 

18.  A  farmer  had  40  sheep  in  one  field,  which  was  | 
of  the  number  in  another  field ;  then  |  of  the  sheep  in 
each  field  jumped  into  the  other;  how  many  then  in  each 
field? 

19.  Maria  gave  away  some  money,  and  then  found  10 
cents,  which  is   ^  of  what  she  then  had,  and  \  of  what 
she  a*,  first  had ;  how  much  did  she  give  away  ? 

20    A  man  lost  |  of  all  his  money,  and  then  won  |  as 


76  MENTAL  ARITHMETIC. 

much  as  lie  lost,  and  then  had  $20     how  much  had  he 
at  first  ? 

21.  If  to  ^  of  the  cost  of  B's  horse  you  add  $80,  the 
sum  will  equal  |  of  the  cost;  required  the  cost  of  his 
horse. 

22.  A  paid  $80  for  flour,  and  f  of  the  number  of  dol- 
lars, is  8  times  the  number  of  barrels  purchased;  what 
was  the  price  of  the  flour  a  barrel  ? 

23.  Philo  found  10  cents,  and  then  lost  |  of  what  he 
found,  and  then  had  j  as  much  as  he  had  at  first;  how 
much  had  he  at  first? 

24.  The  distance  from  Medway  to  Columbia  is  42 
miles,  and  ^  of  this  distance  is  |  of  the  distance  from 
Medway  to  Rockville;  required  the  distance. 

25.  A  has  8  marbles,  and  B  has  7,  and  6  times  what 
they  both  have,  is  equal  to  the  number  that  C  has,  in- 
creased by  10  ;  how  many  has  G  ? 

26.  Mary  has  5  roses,  Jane,  3  times  as  many,  lack- 
ing 5,  and  Susan  has  twice  as  many  as  both,  increased 
by  5 ;  how  many  has  each,  and  how  many  have  they 
all? 

27.  Frescoln  had  20  apples,  and  Lucy  gave  him  10 
more,  he  then  gave  his  father  7,  and  his  mother  a  certain 
nrrnber,  and  had  13  remaining;  how  many  did  he  give 
his  mother  ? 

28.  A  and  B  had  each  30  apples;  A  gave  B  10  of 
his,  and  B   gave  A  6  of  his,  and  then  lost  a  certain 
number,  so  that  A  had  12  more  than  B ;  how  many  did 
B  lose  ? 

29.  A  and  B  had  each  40  cents,  A  gave  B  10  of 
his,  and  B  gave  A  twice  as  many  of  his,  and  then  losing 
a  certain  number,  had  twice  as  many  as  A  gave  him  j 
bow  many  did  he  lose  ? 

30.  The  distance  from  Oak-land  to  Parkesburg  is  16 
miles,  and  |  of  this  distance  is  -*   of  the  distance  from 
Parkesburg  to  Marietta,  lacking  1  mile ;  what  is  the 
distance  ? 


MENTAL  ARITHMETIC.  77 

31.  The  distance  from  Mount  Joy  to  Harrisburg  ia 
25  miles,  and  |  of  this  distance  is  f  of  |  of  the  distance 
from  Harrisburg  to  McVeytown;  what  is  the  distance 
to  McVeytown  ? 

32.  From  Philadelphia  to  Lancaster  the  distance  is 
68  miles,  and  ^  of  this,  increased  by  2  miles,  equals  \ 
of  the  distance  from  Lancaster  to  Harrisburg,  minus  1 
mile ;  required  the  distance  to  Harrisburg. 

We  have  derived  the  principles  of  the  following  propositions 
by  induction  from  analytical  processes ;  we  will  now  proceed  to 
establish  their  .truth  by  rigid  demonstration. 

PROP.  1. — Multiplying  the,  numerator  of  a  fraction  by  any  num- 
ber multiplies  the  value  of  the  fraction  by  that  number. 

PROP.  2. — Dividing  the  numerator  of  a  fraction  by  any  number 
divides  the  value  of  the  fraction  by  that  number. 

PROP.  3. — Multiplying  the  denominator  of  a  fraction  by  any 
number  divides  the  value  of  the  fraction  by  that  number. 

PROP.  4. — Dividing  the  denominator  of  a  fraction  by  any  number 
multiplies  the  value  of  the  fraction  by  that  number. 

PROP.  5. — Multiplying  both  numerator  and  denominator  of  a 
fraction  by  any  number  does  not  change  the  value  of  the  fraction. 

PROP.  6. — Dividing  both  numerator  and  denominator  of  a  frac- 
tion by  any  number  does  not  change  the  value  of  the  fraction. 

DEMONSTRATION  OF  THE  FIRST. — If  we  multiply  the  numera- 
tor of  a  fraction  by  any  number,  as  6,  the  resulting  fraction 
will  express  5  times  as  many  parts  each  of  the  same  size  as 
before;  hence  the  value  of  the  fraction  is  5  times  as  great. 

DEMONSTRATION  OF  THE  THIRD. — Since  the  denominator  showa 
the  number  of  equal  parts  into  which  the  unit  is  divided,  if  we 
multiply  the  denominator  by  any  number,  as  4,  the  unit  will  be 
divided  into  4  times  as  many  parts,  hence  each  part  will  be  \ 
as  large  as  before,  and  the  same  number  of  parts  being  taken 
the  value  of  the  fraction  will  be  \  as  great. 

DEMONSTRTAION  OF  THE  FIFTH. — Since  multiplying  the  nume- 
rator multiplies  the  value,  and  multiplying  the  denominator 
divides  the  value  of  the  fraction,  multiplying  both  numerator 
and  denominator  by  the  same  number,  multiplies  and  divides 
the  value  by  the  same  number,  and  hence  does  not  change  its 
value.  Therefore,  £c. 

NOTE. — The  2d  is  demonstrated  very  much  like  the  1st,  the 
4th  like  the  3d,  the  6th  like  the  5th. 


78  MENTAL  ARITHMETIC 


SECTION    IV. 

LESSON  I. 
FEDERAL  MONEY. 

10  mills  (m.)     .     ,  equal  1  cent,  .     ,     «  ct. 

10  cents        ...  "     1  dime,  j     ,  d. 

10  dimes       ...  "     1  dollar,  .     .     c  9 

10  dollars      ...  "1  eagle,  ...  E. 

Federal  money  is  the  currency  of  the  United  StAtes.  The 
coins  are  of  three  kinds,  gold,  silver,  and  copper.  The  new  cent 
coin  consists  of  88  parts  copper  and  12  parts  nickle.  The  gold 
and  silver  coins  contain  JL  alloy,  excepting  the  3  cent  piece, 
which  is  1  alloy.  The  alloy  of  the  silver  coin  consists  of  copper, 
and  of  the  gold  coin,  of  equal  parts  of  copper  and  silver. 

Dollars  and  cents,  written  together,  are  separated  by  a  point  (.) , 
thus  $5.60  is  read  5  dollars  and  60 'cents. 

1.  How  many  mills  in  4  cents?     In  7  cents?     In  8 
dimes  ? 

2.  How  many  cents  in  5  dimes?     In  6  dollars?     In  3 
eagles  ? 

3.  How  many  dollars  in  9  eagles  ?     In  40  dimes  ?     In 
500  cents  ? 

4.  How  many  eagles  in  50  dollars  ?     In  300  dimes  ? 
In  7000  cents  ? 

5.  If  5  apples  cost  20  cents,  how  many  apples  can  you 
buy  for  $2  ? 

6.  How  many  sheep  can  you  b~iy  for  18  eagles,  at  the 
rate  of  3  sheep  for  15  dollars  ? 

7.  What  part  of  2  dollars  is  5  cents,  and  what  part  of 
3  dimes  is  6  cents  ? 

8.  What  part  of  4  eagles  is  8  dimes,  and  what  part  of 
5  cents  is  |  of  a  dime  ? 


MENTAL  ARITHMETIC.  79 

ENGLISH  OR  STERLING  MONEY. 

4  farthings  (gr.)     .     .  equal  1  penny,    .  d. 

12  pence "     1  shilling,  .     s. 

20  shillings     ....        :"     1  pound,    .  .     £ 

21  shillings     ....         "1  guinea. 

1  £  =  $4.84.  1  s.  =  $0.24.  1  d.  =  $0.02. 

The  symbols  £,  s.,  d.,  and  qr.  are  the  initials  of  the  Latin  words 
libra,  solidus,  denarius,  and  quadrans ;  signifying,  respectively 
pound,  shilling,  penny,  and  quarter. 

1.  How  many  farthings  in  2  pence?     In  3?     In  5? 
In  6?     In  8?     In  10?  ^ 

2.  How  many  pence  in  8  farthings?     In  2?     In  20? 
In  28?     In  30'? 

3.  How  many  pence  in  2  shillings?     In  4?     In  5? 
In  6?     In  10? 

4.  How  many  shillings  in  24  pence?    In  48?    In  72? 
In  96?     In  150? 

5.  How  many  shillings  in  3  pounds?     In  5?     In  6? 
In  8?     In  12? 

6.  How  many  pence  in  3s.  and  6d.  ?  In  1£,  2s.  and  4c?.  ? 

7.  What  part  of  2  pence  is  3  farthings,  and  what  pait 
of  3  shillings  is  5  pence? 

8.  What  part  of  8  pence  is  f  of  a  penny,  and  what 
part  of  a  guinea  is  |  of  a  pound  ? 


LESSON  II 


TROY  WEIGHT. 

24  grains  (#r.)     .     .     equal  1  pennyweight,     pw*,. 
20 'pennyweights       .         "     1  ounce,      .     .     oz. 
12  ounces  ....         "1  pound,     .     .     Ib 

The  term  Troy  is  said  to 'be  derived  from  Troyes,  the  name  of 
a  town  in  France,  where  the  weight  was  first  used  in  Europe. 
The  symbol  (oz.)  is  from  the  Spanish  word  onza^  for  ounce,  and 
(Ib.)  from  libra,  a  pound. 


80  MENTAL  ARITHMETIC. 

1.  How  many  grains  in  2  pwts.  ?     In  3  »      In  4  ?     In 
5?     In  6?     In  7? 

2.  How  many  pwts,  in  48  grs.  ?    In  72  ?    In  96?    ID 
120?     In  240? 

3    How  ,many  pwts.  in  3  oz.?     In  4?     In  5?     In  6? 
la  7?     In  10? 

4.  How  many  ounces  in  5  pounds?     In  7?     In  10? 
In  8?     In  4?     In  12? 

5.  If  10  pwts.  of  silver  are  worth  3  shillings,  what  is 
the  value  of  5  Ibs.  of  silver  ? 

6.  What  is  the  value  of  3  ounces  of  gold,  if  3  grains 
are  worth  3  dollars  ? 


APOTHECARIES'  WEIGHT. 

20  grains  (gr.~)  .  .  equal  1  scruple,  .  .  9 

3  scruples  ...  u  1  dram,  .  .  .  3 

8  drams  ....  "1  ounce,  .  '.  .  3 

12  ounces  ....         "1  pound,  .  .  .  Ib 

This  weight  is  used  for  mixing  and  retailing  medicines.    The 
pound  is  the  same  as  the  pound  Troy. 

1.  How  many  grs.  in  3  9s?    In  5  ^s?    In  2  gs?    In 

ift? 

2.  How  many  scruples  in  4  ^s?     In  40  grs.?     In  2 
§s  and  3  £s? 

3.  How  many  ounces  in  3  Ibs?     In  16  £s?     In  4  Iba 
and  5  gs? 

4.  How  many  drams  in  120  grs.?     In  36  9s?     In  3 
pounds  ? 

5.  Which  is  the  heavier,  an  ounce  of  opium  or  an  ounce 
of  silver  ? 

6.  If  5  grs.  of  medicine  cost  10  cents,  what  will  3  oz. 
and  4  gs  cost  ? 

7.  Two-thirds  of  9  scruples  of  a  certain  drug  cost  18 
cents,  what  will  3  fourths  of  8  pounds  cost  ? 


MENTAL  ARITHMETIC.  81 

AVOIRDUPOIS  WEIGHT. 

16  drams  (<#r.)     .     .  equal  1  ounce,       .     .        oz. 

16  ounces  ....  u     1  pound,      .     .        Ib. 

25  pounds  ....  "1  quarter,    .     .        qr. 

4  quarters      ...  "     1  hundred  weight,  cwt. 

20  hundred  weight    .  "     i  ton,     ...        T. 

The  term  Avoirdupois  is  derived  from  the  French  avoir  du 
poids,  signifying  to  have  weight.  The  pound  consists  of  7000 
Troy  grains.  This  weight  is  used  for  weighing  almost  all 
articles  except  gold,  silver,  platina,  and  precious  stones,  which 
are  weighed  by  Troy  Weight. 

1.  How  many  drams   in  2  ounces?     In  3  ?     In  5? 
In  10? 

2.  How  many  ounces  in  3  pounds  ?     In  5  ?     In  48 
drams  ? 

3.  How  many  quarters  in  75  pounds?     In  3200  oz.  ? 
In  5  cwt.  ? 

4.  How  many  hundredweight  in  36  quarters  ?    In  300 
Ibs.  ?     In  6  tons  ? 

5.  How  many  tons  in  5  hundredweight?    In  240  qrs. ? 
In  600  Ibs.  ? 

6.  What  will  12  pounds  of  starch  cost,  if  o  ounces  cost 
20  cents  ? 

7.  Wha,t  will  2  cwt.  of  coffee  cost,  at  the  rate  of  4 
pounds  for  60  cents  ? 

8.  I  gave  3  cwt.  2  qrs.  of  hay,  worth  $20  a  ton,  for 
butter  worth  25  cents  a  pound;  how  many  pounds  of 
butter  did  I  receive  ? 

9.  Which  is  the  heavier,  a  pound  of  gold  or  a  pound 
of  lead  ?     An  ounce  of  silver  or  an  ounce  of  feathers  ? 

METRIC  SYSTEM. — In  the  Metric  System  of  weights  and 
measures  the  unit  of  weight  is  the  Gram.  The  gram  =  15.44 
Troy  grs.  ;  it  is  used  for  light  weights.  The  Kilogram  (1000 
grams),  abbreviated  into  kilo,  will  be  the  ordinary  weight  of 
commerce.  The  kilogram  —  2£  Ibs.  Avoirdupois,  very  nearly. 


82  MENTAL  ARITHMETIC 


LESSON  III. 
LONG  MEASURE. 

12  inches  (in.)    .     .  equal  1  foot,     ,  .  ft. 

3  feet       ....  "1  yard,    .  ,  yd. 

5i  yards  ....  "     1  rod,      .  .  .  rd. 

40~rods      ....  "     1  furlong,  .  .  fur. 

8  furlongs     .   - .     .  *     "     1  mile,    .  .  .  m. 

The  yard  is  the  standard  unit  of  length.  It  is  formed  by 
dividing  a  pendulum,  which  vibrates  seconds  in  a  vacuum,  at 
the  level  of  the  s«a,  in  the  latitude  of  London,  into  391393 
equal  parts  and  taking  360000  of  these  parts.  From  this  unit 
all  other  measures  and  weights  are  derived. 

1.  How  many  inches  in  3  feet?     In  5  ?     In  7  ?     In 
2  yds.?     In  2  rods  and  2  feet? 

2.  How  many  feet  in  60  inches?     In  96?     In  108? 
In  6  rds.  ?     In  4  rds.  7  yds.  ? 

3.  How  many  yards  in  45  feet?     In  66?     In  72  in.? 
In  4  rds.  ?     In  1  fur.  4  rods  ? 

4.  How  many  furlongs  in  320  rods?     In  440?     In 
660  feet  ?     In  9  miles  ? 

5.  How  many  miles  in*  104  furlongs?     In  640?     In 
800?     In  1760  yards? 

6.  Mary  ran   60  rods,  and  4  fifths  of  this  distance 
equals  T3<j  of  the  distance  Henry  ran ;  how  many  furlongs 
did  Henry  run  ? 

7.  Three  fourths  of  the  length  of  a  pole  is  6  feet,  which 
is  2  sixths  of  the  length  of  another  pole ;  how  many  yards 
long  is  each  pole  ? 

8.  What  part  of  1  yard  is  2  feet,  and  what  part  of  2 
furlongs  is  4  yards? 

METRIC  SYSTEM. — In  the  Metric  System  the  unit  of  length  is 
the  Meter.  It  equals  39.37  inches,  or  very  nearly  3  ft.  3  in.  and 
|  of  an  inch.  Long  distances  will  be  measured  by  the  Kilometer 
( ]  000  meters),  which  equals  f  of  a  mile,  very  nearly. 


MENTAL  ARITHMETIC.  83 

CLOTH  MEASURE, 

2J  inches      .     .     .     equal  1  nail,     ,     .          na. 
4  nails      ....         "     1  quarter,    .  qr. 

4  quarters     ...         "     1  yard,    .     .     .     yd. 

8  quarters     ..."     1  Ell  Flemish, .     E.  Fl 

5  quarters     ...         "     1  Ell  English,  .     E.  E. 

6  quarters     ...         "     1  Ell  French,   .     E.  Fr 

Cloth  Measure  is  used  in  measuring  cloth,  lace,  muslin,  &c. 

1.  How  many  inches  in  4  nails?     In  5  quarters?     In 

3  ells  Flemish?     In  2  ells  English? 

2.  How  many  nails  in  18  inches?    In  54  inches?    In 

4  E.  E.  and  3  qrs.  ?     In  5  E.  Fr.  and  2  qrs.  ? 

3.  How  many  quarters  in  24  nails?     In  63  inches? 
In  3  E.  E.,  4  E.  Fl.,  and  2  qrs.  ?     In  52  yards  ? 

4.  How  many  yards  in  144  inches,  48  nails,  and  32 
qrs.  ?     In  7  E.  Fr.  and  2  qrs.  ? 

5.  What  is  the  difference  between  8  E.  Fr.  and  4  E. 
E.?     Between  5  E.  Fl.  and  32  nails? 

6.  How  many  coats  can  be  made  from  9  E.  E.  and  3 
qrs.,  if  one  coat  require  3  yards  ? 

7.  Four  fifths  of  the  number  of  yards  in  Mary's  dres? 
equals  8,  which  is  4  times  |  as  many  yards  as  her  cloak 
contains;  how  many  yards  more  in  the  dress  than  cloak; 


LESSON  IV. 


ALE  OR  BEER  MEASURE. 

2  pints  (p£.)     .     .     equal  1  quart,  .     ,  .  qt. 

4  quarts  ...  "  1  gallon,  .  .  gal. 

36  gallons  ...  "  1  barrel,  .  .  bar. 

54  gallons  ...  "1  hogshead,  .  lihd. 

Ale  or  Beer  Measure  is  used  in  measuring  ale,  beer,  &c.    Tha 
gallon  consists  of  282  cubic  inches. 


84  MENTAL  ARITHMETIC. 

1.  How  many  pints  in  5  quarts?    In  10  gallons?     In 
5  barrels  ?     In  |  of  a  hogshead  ? 

2.  How  many  quarts  in  46  pints?     In  7  gallons?     In 
3  bar.  10  gals.  ?     In  2  hhd.  2  gals.  3  qts.  ? 

3.  How  many  gallons  in  64  pints  and  20  quarts?     In 
|  of  a  hhd.  and  |  of  a  barrel? 

4.  What  part  of  a  hhd.  is  J  of  a  barrel  ?    What  is  the 
difference  between  2  hhd.  and  3  barrels  ? 

5.  What  part  of  2  barrels  is  9  gallons  ?     What  cost  a 
gal.  of  aie;  at  5  cents  a  pint  ? 

LIQUID  OR  WINE  MEASURE. 

4  gills  ((71.)     .     .     .     equal  1  pint,  .  .  pt. 

2  pints        ....  "     1  quart,  .  .  qt. 

4  quarts  ....  "  1  gallon,  .  .  gal. 

31£  gallons  ...  "1  barrel,  .  .  bar. 

42  gallons  ...  "1  tierce,  .  .  tier. 

63  gallons  ...  "1  hogshead,  .  hhd. 

2  hogsheads    ...  "     1  pipe,  .  .  pi. 

2  pipes       ....  "     1  tun,  .  .  tun. 

Wine  Measure  is  used  in  measuring  wine,  and  liquids  gene- 
rally.    The  gallon  consists  of  231  cubic  inches. 

1.  How  many  gills  in  6  pints?     In  5  quarts?     In  8 
gallons  ?     In  2  barrels  ? 

2.  How  many  pints  in  24  gills?     In  8  quarts?     In  2 
tierces  ?     In  1  hhd.  and  7  gals.  ? 

3.  How  many  quarts  in  40  gills?     In  26  pints?     In 
2  tuns  and  3  hogsheads? 

4.  What  part  of  §  of  a  hogshead  is  ^  of  a  tierce  ?   What 
part  of  |  of  a  hhd.  is  7  quarts  ? 

5.  What  part  of  ^  of  a  gal.,   wine  measure,  is  |  of  a 
gal.,  beer  measure  ? 

6    Which  is  the  greater,  a  gallon  of  wine  or  a  gallon  of 
beer?     A  hogshead  of  ale  or  molasses? 

METRIC  Si  STEM.— The  unit  of  capacity  is  the  Liter,  equal  to 
lT'g  liquid  quarts,  very  nearly.  The  Hectoliter  (100  liters), 
equal  *o  about  2|  bus.,  will  be  used  for  measuring  grain,  etc. 


MENTAL  ARITHMETIC.  85 

DRY  MEASURE. 

2  pints  (p£.)    .     .     •  equal  1  quart,  .  qt* 

8  quarts      ....         "     1  peck,  .  pk. 

4  pecks      ....        "     1  bushel,  bu. 

36  bushels    ....         "1  chaldron,  .  ch. 

Dry  Measure  is  used  for  measuring  grain,  fruit,  coal,  &c. 

1.  How  many  pints  in  3  quarts  ?    In  2  pecks?    In  2.| 
bushels  ?     In  one  chaldron  ? 

2.  How  many  quarts  in  64  pecks?     In  10  bushels? 
In  |  of  a  chaldron  ?     In  24  pints  ? 

3.  How  many  pecks  in  5^  bushels  and  3  pecks?     In 
96  pints  ?     In  56  quarts  ? 

4.  How  many  bushels  in  2  ch.  8  bu. ?   In  192  quarts? 
In  128  pints?     In  78  pecks? 

5.  Which  cost  the  more,  and  how  much,  5  bu.  3  qts. 
of  salt,  at  4  cents  a  quart,  or  10  bu.  3  pks.  of  apples,  at 
50  cents  per  bushel  ? 

6.  At  ten  cents  a  peck,  how  many  bushels  of  pears  can 
be  bought  for  8  dollars  ? 

7.  A  grocer  bought  7  bushels  and  2  pecks  of  cherries 
at  the  rate  of  3  cts.  a  pint,  and  sold  them  for  2  cwt.  2 
qrs.  of  sugar  at  6  cts.  per  Ib. }  how  much  did  he  gain  ? 


LESSON  V. 

TIME. 

60  seconds  (sec.)  .     .     equal  1  minute,  .  .     ra. 

60  minutes       ...         "     1  hour,  .  .  .     hr. 

24  hours      ....         "1  day,     .  .          da. 

7  days        ....         "1  week,  .  .  .     wJc. 

4  weeks     ....         "     1  month,  .  .     mo. 

52  weeks     ....         "     1  year,    .  .  .     yr. 

12  calendar  months    .         "     1  year,   .  .  .     yr, 
8* 


86  MENTAL  ARITHMETIC. 

1.  How  many  seconds  in  2  minutes?     In  3  ?     In  6? 
In  8?     In  10? 

2.  How  many  minutes  in  3  hours?     In  4  ?     In  5? 
In  120  seconds? 

3.  How  many  hours  in  4  days?     In  5?     In  10?     In 
240  minutes? 

4.  How  many  days  in  3  weeks  ?     In  9  ?     In  12  ?     In 
96  hours? 

5.  How  many  weeks  in  7  months  ?     In  11  ?     In  28 
days?     In  168  hours? 

6.  How  many  months  in  10  yearj  ?   In  24  weeks  ?   In 
56  days?     In  112  days? 

7.  How  many  years  in  24  months?     In  104  weeks? 
In  364  days? 

8.  Mary  is  4  years  and  6  months  old,  and  her  brother 
is  6  years  and  4  months  old ;  how  much  younger  is  Mary 
than  her  brother  ? 

9.  How  many  days,  of  10  hours,  each,  will  he  required 
to  make  40  hats,  if  4  hats  can  be  made  in  7  hours  ? 

CHANGE  OF  CURRENCIES. 

In  New  England,  Virginia,  Kentucky,  and  Ten- 
nessee,   6*.  =$1. 

In  New  York,  Ohio,  and  North  Carolina,      .     .  8*'.        =  $1 

In  New  Jersey,  Pennsylvania,  Delaware,  and 

Maryland, 7s.6e?.  =  $l. 

In  South  Carolina  and  Georgia, 4s  Sd.  =$1. 

In  Canada  and  Nova  Scotia, 5s.        =$1. 

1.  Ten  dollars,  in  New  York,  equal  how  many  pounds? 

SOLUTION. — In  $1  there  are  8  shillings,  and  in  £1  there  arc 
20  shilling  One  shilling  is  1  of  a  pound,  and  8  shillings, 
or  $1,  are  8  times  ^1  ,  which  are  ^  or  £|,  and  $10  are  10  times 
|,  wh  ch  are  £4.  ~  Therefore,  $10,  in  New  York,  equals  £4. 

2.  How  many  pounds  in  $20  in  Ohio?     In  $60  in 
North  Carolina? 

3.  How  many  pounds  in  $30  of  New  England  ?     ID 
$90,  Kentucky  ?     In  $1 00,  Virginia  ? 


MENTAL  ARITHMETIC.  87 

4.  How  manj  pounds  in  $48,  New  Jersey  I     In  $64, 
Pennsylvania?     In  $72,  Maryland? 

5.  How  many  pounds  in  $120,  South  Carolina?     In 
8240,  Georgia  ?     In  $56,  Canada  ? 

6.  How  many  dollars  in  £24,  New  York?     In  £26, 
Ohio  ?     In  £30,  North  Carolina  ? 

7.  How  many  dollars  in  £27,  Maine?     In  £54,  Ver- 
mont ?     In  £60,  Massachusetts  ? 

8.  How  many  dollars  in  £21,  South  Carolina?     In 
£63,  Georgia  currency  ? 

9.  What  is  the  value  of  a  shilling,  in  cents,  in  each 
of  the  states  mentioned  in  the  table  ? 

REMARK.— The  class  should  now  review,  unless  they  are 
entirely  familiar  with  the  preceding  sections. 


SECTION   Y. 

LESSON  I. 

Special  attention  is  called  to  the  adaptation  of  the  remainder 
of  the  book  to  elementary  and  advanced  classes.  Each  lesson 
commences  simply  enough  for  the  most  elementary  pupils,  and 
near  the  close  becomes  sufficiently  difficult  for  the  most  ad- 
vanced. The  more  difficult  problems  are  separated  from  the 
easier  by  a  line ;  these  are  to  be  omitted  by  young  pupils  until 
review,  while  the  older  or  more  advanced  pupils  take  the  whole 
lesson. 

1.  If  |  of  a  yard  of  cloth  cost  J  of  a  dollar,  what  will 
|  of  a  yard  cost  ? 

2.  If  8  horses  eat  a  quantity  of  hay  in  16  weeks,  how 
long  would  it  last  32  horses? 

3.  If  5  men  earn  30  dollars  in  a  certain  time,  how 
much  will  8  men  earn  in  j  the  time  ? 

4.  If  6  persons  spend  836  in  3  days,  how  much  wiD 
10  persons  spend  in  5  days  ? 


88  MENTAL  ARITHMETIC. 

5.  How  long  will  5  tons  of  hay  last  8  horses,  if  6  horses 
eat  it  in  12  weeks  ? 

6.  How  long  will  3  barrels  of  flour  last  10  persons,  if 
4.  persons  eat  4  barrels  in  40  weeks  ? 

7.  If  7  men  can  earn  82 8  in  4  days,  how  many  dollars 
can  9  men  earn  in  6  days  ? 

8.  How  long  will  6  men  require  to  build  6  boats,  if  7 
men  build  3  boats  in  12  weeks  ? 

9.  If  10  oxen  eat  4  acres  of  grass  in  6  days,  in  how 
many  days  will  30  oxen  eat  .8  acres  ? 

10.  If  it  require  4  men  7  days  to  perform  a  certain 
piece  of  work,  how  many  men  can  perform  a  piece,  3 
times  as  large,  in  6  days'/ 

11.  If  it  require  5  men  8  days  to  build  20  rods  of 
wall,  how  many  men  can,  in  2  days,  build  |  as  much 
wall? 

12.  How  many  men,  in  10  days  of  6  hours  each,  can 
earn  as  much  as  6  men  in  20  days  of  8  hours  each  ? 

13.  How  many  oxen  will  eat  5  tons  of  hay  in  5  weeks, 
if  12  oxen  eat  4  tons  in  4  weeks  ? 

14.  If  3  horses,  in  J-  of  a  month,  eat  |  of  a  ton  of  hay, 
how  long  will  |  of  a  ton  last  5  horses  ? 

15.  If  4  men  can  do  a  piece  of  work  in  6  days,  in  what 
time  will  it  be  completed  if  they  receive  the  assistance  of 
5  men,  when  the  work  is  half  done  ? 

16.  How  many  cents  are  10  melons  worth,  if  4  melona 
are  worth  8  oranges,  and  3  oranges  are  worth  9  cents  ? 

17.  How  many  cents  will  5  oranges  cost,  if  3  oranges 
are  worth  9  apples,  and  4  apples  are  worth  8  cents  ? 

18.  How  many  dollars  will  10  sheep  cost,  if  5  sheep 
are  worth  2  cows,*  and  4  cows  are  worth  180  ? 

19.  How  many  pigs  can  a  man  get  for  2  cows,  if  12 
pigs  are  worth  3  sheep,  and  12  sheep  are  worth  3  cows  ? 

20.  How  many  oranges  can  you  buy  for  20  cents,  of  4 
oranges  are  worth  8  apples,  and  4  apples  are  worth  8 
cents? 

21     How  many  hens  can  you  purchase  for  $12,  if  4 
hens  are  worth  8  turkeys,  and  3  turkeys  are  worth  $6  ? 


MENTAL  ARITHMETIC.  89 

22.  If  G  sheep  are  worth  2  cows,  and  10  cows  are 
worth  5  horses,  how  many  sheep  can  you  buy  for  3 
horses  ? 

23.  If  a  measure  of  flour  make  5  four  cent  loaves,  how 
many  2  cent  loaves  will  it  make  ?     How  many  5  cent 
loaves  will  it  make  ? 

24.  If  a  certain  sum  of  money  buy  9  four  cent  oranges, 
how  many  6  cent  oranges  can  you  buy  with  the  same 
gum?  ' 

25.  If  a  5  cent  loaf  weigh  7  ounces,  when  flour  is 
worth  6  dollars  a  barrel,  how  much  should  it  weigh  when 
flour  is  worth  7  dollars  per  barrel  ? 

26.  If  a  3  cent  loaf  weigh  9  ounces,  when  flour  is  6 
dollars  a  barrel,  how  much  ought  a  4  cent  loaf  to  weigh 
when  flour  is  $S  a  barrel  ? 


27.  A  can  do  as  much  work  in  2  days  as  B  can  in  4, 
or  C  in  6  days ;  in  how  many  days  can  B  do  as  much  as 
C  can  in  18  days  ? 

28.  A  can  do  as  much  in  6  days  as  B  can  in  2  days, 
and  B  can  do  as  much  in  5  days  as  C  can  in  15  days; 
in  how  many  days  can  A  do  as  much  as  C  can  in  4  days  ? 

29.  A  can  do  3  times  as  much  in  a  day  as  B,  and  B 
can  do  twice  as  much  as  C ;  in  how  many  days  can  A  do 
as  much  as  C  can  in  4  days  ? 

30.  If  5  horses  can  eat  a  lot  of  grain  in  12  days,  in 
what  time  will   it  be   consumed,  if  7  horses  are  added 
when  the  grain  is  |  eaten  ? 

31.  If  8  boys  can  weed  a  garden  in  5  hours,  in  what 
time  will  the  job  be  completed,  provided  3  boys  leave 
when  the  work  is  half  done  ? 

32.  If  9  men  build  10  rods  of  wall  in  8  days,  in  what 
time  can  20  rods  be  built,  if  f  of  their  number  leave 
when  the  work  is  \  part  completed  ? 


90  MENTAL  ARITHMETIC. 


LESSON  II. 

1.  A  gentleman  gave  4  cents  each  to  some  poor  child- 
ren ;  had  he  given  them  7  cents  each  it  would  have  taken 
B6  cents  more ;  how  many  children  were  there  ? 

SOLUTION. — By  the  second  condition  of  the  question  fre  gave 
each  child  7  — 4,  which  is  3  cents  more  than  by  the  first,  and 
to  them  all,  36  cents  more ;  hence  there  were  as  many  children 
as  3  is  contained  times  in  36,  which  are  12.  Therefore,  &c. 

2.  A  teacher  gave  his  pupils  2  questions  each,  and  had 
26  questions  remaining;  if  he  had  given  them  4  apiece 
there  would   have  been  none    remaining;    required  the 
number  of  pupils  and  questions. 

3.  A  father  gave  his  sons  5  dollars  each,  and  had  30 
dollars  remaining;  had  he  given  them  8  dollars  each  it 
would  have  taken  all  his  money ;  required  the  number  of 
eons  and  amount  of  money. 

4.  Mary  gave  some  beggars  6  cents  each,  and  had  25 
cents  remaining;  had  she  given  them  8  cents  each  she 
would  have  had  3  cents  remaining ;  how  many  beggars 
were  there  ? 

5.  Edward  bought  a  certain  number  of  melons,  at  the 
rate  of  5  cents  each;  if  he  had  paid  3  cents  each  they 
would  have  cost  14  cents  less ;  how  many  melons  did  he 
buy? 

6.  A  lady,  wishing  to  buy  some  ribbon,  found  if  she 
bought  that  at  10  cents  a  yard  she  would  want  9  cents  to 
pay  for  it,  but  if  she  bought  that  at  7  cents  ji  yard  she 
would  have  9  cents  remaining;  how  much  money  had 
ihe? 

7.  Morris  and  Robert  have  each  a  certain  number  of 
peaches ;  if  Morris  had  10  more  he  would  have  twice  as 
many  as  Robert,  but  if  he  had  30  more  he  would  have  4 
times  as  many  as  Robert;  how  many  has  each? 

8.  A  drover  bought  a  number  of  sheep  at  $3-J  a  head, 
and  found  he  lacked  $6  of  having  money  enough  to  pay 


MENTAL  ARITHMETIC.  91 

for  them ;  if  he  had  paid  $2  a  head  he  would  have  had  $9 
remaining ;  how  much  money  had  he  ? 

9.  Sallie  wishes  to  buy  a  silk  dress;  if  she  pays  $5  a 
yard  she  will  lack  $10  of  having  money  enough  to  pay 
for  it,  but  if  she  pays  $2.50  a  yard  she  will  have  Sf)  re- 
maining; required  her  money,  and  the  number  of  yards 
in  the  dress  ? 

10.  A  certain  number  of  oranges,  at  the  rate  of  3  for 
12  cents,  will  cost  18 'cents  more  than  the  same  number 
of  apples,  at  the  rate  of  4  for  8  cents;   required  the 
number. 

11.  A  certain  number  of  peaches,  at  3  for  10  cents, 
will  cost  20  cents  more  than  the  same  number  of  pears, 
at  the  rate  of  3  for  5  cents ;  required  the  number. 

12.  James    and    Henry   have    a   certain    number   of 
marbles ;  if  James  had  8  more  he  would  have  6  times 
as  many  as  Henry,  but  if  12  less  he  would  have  only 
twice  as  many ;  how  many  has  each  ? 

13.  A  gentleman  divided  28  apples  between  an  equal 
number  of  boys  and  girls,  giving  to  each  girl  3,  and  to 
each  boy  4  apples ;  required  the  number  of  boys  and  girls. 

14.  A  man  bought  an  equal  number  of  pigs  and  sheep 
for  $81,  giving  $4  each  for  the  pigs,  and  $5  each  for  the 
sheep ;  how  many  of  each  did  he  buy  ? 

15.  A  boy  expended  36  cents  for  an  equal  number  of 
melons  and  lemons,  giving  4  cents  each  for  the  melons, 
and  2  cents  each  for  the  lemons ;  how  many  of  each  did 
he  purchase  ? 


16.  A  lady  gave  60  cents  to  some  poor  children ;  to 
each  boy  she  gave  2  cents,  and  to  each  girl  4  cents;  how 
many  were  there  of  each,  provided  there  were  3  times  as 
many  boys  as  girls? 

17.  Two  boys  had  an  equal  sum  of  money;  one  bought 
a  certain  number  of  oranges,  at  4  cents  each,  and  had  12 
cents  remaining  •  the  other  bought  twice  as  many  apples. 


92  MENTAL  ARITHMETIC. 

for  3  cents  each,  and  had  2  cents  remaining;  how  much 
money  had  each  ? 

18.  A,  B,  and  C,  dig  a  ditch  for  $60 ;  A  receives  $H. 
B  $2,  and  C  $2-|  a  day ;  how  many  days  were  they  em- 
ployed, and  what  did  each  receive  ? 

19.  A  and.  B  agree  to  perform  a  piece  of  work,  A  re* 
ceiving  $2,  and  B  $3  a  day ;  A  works  twice  as  many  days 
as  B,  and  they  together  receive  870 ;  how  many  days  did 
each  labour? 


LESSON  IIL 

1.  If  to  Harry's  age  its  ^  be  added,  the  sum  will  be 
24  years ;  what  is  his  age  ? 

SOLUTION. — By  the  condition  of  the  problem,  2  Of  Harry's  age, 
plus  £  of  his  age,  which  is  ^  of  his  age,  equals"  24  years.  If  3 
of  Harry's  age  equals  24  years,  .1  of  his  age  equals  1  of  24  years', 
which  is  8  years,  and  £,  or  his  age,  equals  2  times  8,  or  16  years. 
Therefore,  &c. 

2.  What  number  is  that  to  which,  if  its  ^  be  *dded, 
the  sum  will  be  36  ? 

3.  Required  the  number,  which  being  increased   by 
its  I  equals  40. 

4.  What  number  is  that,  which  being  increased  by  its 
£  the  sum  will  be  80  ? 

5.  What  number  is  that,  which  being  diminished  by 
its  |  the  remainder  will  be  30  ? 

6.  Three  times  a  certain  number,  increased  by  f  of 
itself,  equals  22 ;  required  the  number. 

7.  Reuben's  age,  being  doubled  and  diminished  by  | 
of  his  age,  equals  50  years;  how  old  is  he? 

8.  Three  and  ^  times  a  number,  minus  f  of  the  num- 
ber, equals  34 ;  what  is  the  number  ? 

9.  Two-fifths  of  a  number  being  subtracted  from  £  of 
the  nurnbe~  equals  7 ;  required  the  Dumber. 


MENTAL  ARITHMETIC.  93 

10  A  boy  being  asked  his  age,  replied,  that  his  aget 
increased  by  its  |  and  -f ,  equalled  39  /ears;  what  was  his 
age  I 

11.  What  number  is  that,  which  being  increased  by 
the  difference  between  its  |  and  i,  equals  42  ? 

12.  If  the  height  of  a  tree  be  increased  by  its  f  and 
10  feet  more,  the  sum  will  be  twice  the  height;  what  is 
the  height  of  the  tree  ? 

13.  Jf  twice  the  length  of  a  pole  be  increased  by  its  3 
and  2  feet  more,  the  sum  will  equal  3  times  the  length 
of  the  pole;  required  its  length. 

14.  If  3  times  Henry's  age  be  increased  by  its  ^,  ^ 
and  2  years  more,  the  sum  will  equal  4  times  his  age, 
what  is  his  age  ? 

15.  If  the  height  of  a  steeple  be  increased  by  its  J, 
and  that  sum  diminished  by  the  difference  between  |  and 
J  of  the  sum,  it  will  equal  J  of  its  height,  minus  13  feet; 
required  its  height. 

16.  Two  times  a  number,  plus  6,  equals  3  times  the 
same  number,  plus  2 ;  what  is  the  number  ? 

17.  Three  times  a  certain  number,  plus  8,  equals  4 
times  the  same  number,  plus  3 ;  required  the  number. 

18.  ^  of  a  certain  number,  increased  by  10,  equals  | 
of  the  same  number,  plus  8 ;  what  is  the  number  ? 

19.  Four  times  A's  age,  diminished  by  10  years,  equals 
3  times  his  age,  increased  by  10  years ;  what  is  his  age  ? 

20.  Two-thirds  of  Morton's   apples,  increased   by  2, 
equals  |  of  his  number,  diminished  by  one ;  how  many 
apples  has  he  ? 

21.  Benton  lost  |  of  all  his  money,  and  then  found  -| 
as  much  as  he  lost,  and  then  had  $120;  how  much  money 
had  he  at  first? 

22.  Mary  gave  |  of  her  money  to  the  poor,  and  then 
found  |  as  much  as  she  gave  away,  and  then  had  $30 ; 
how  much  had  she  at  first? 

23.  William  borrowed  f  of  Emily's  money,  and  after 
spending  |  of  it,  returned  the  remainder,  which  was  $20  ; 
how  much  money  had  Emily? 


94  MENTAL  ARITHMETIC. 

24.  A  thief  stole  J  of  Harry's  money,  and  before  he 
was  caught  spent  f  of  it ;  the  remainder,  which  was  $20 
ess  than  he  stole,  was  given  back ;  how  much  money  had 

Harry  ? 

25.  Two  times  a  certain  number,  -f- 10,  equals  3  limes 
the  sum  obtained  by  increasing  the  number  by  2 ;  what 
is  the  number?  i 

26.  Baldwin  had  stolen  from  him  |>-  of  his  money,  arid 
the  thief  was  not  caught  until  he  had  spent  4  of  it ;  the 
remainder,  which  was  $30  less  than  Baldwin  had  remain- 
ing, was  given  back ;  how  much  money  had  Baldwin '/ 


LESSON  IV. 

1.  William  and  Henry  have  15  marbles;  how  many 
has  each,  provided  William  has  twice  as  many  as  Henry? 

SOLUTION. — By  a  condition  of  the  problem,  twice  Henry's 
number  equals  William's,  which,  added  to  Henry's  number, 
equals  three  times  Henry's,  which  is  what  they  both  have,  or  15 
marbles.  If  3  times  Henry's  number  equals  15,  once  his  num- 
ber equals  J  of  15,  which  is  5,  and  twice  his  number,  or  Wil- 
liam's, equals  twice  5,  or  10  marbles.  Therefore,  &c. 

2.  Robert  has  3  times  as  many  cents  as  Elia-s,  and  they 
together  have  24 ;  how  many  has  each  ? 

3.  William  has  4  times  as  many  nuts  as  Oliver,  and 
they  together  have  20  pints;  how  many  pints  has  each  ? 

4.  Emma  has  35  flowers,  and  4  times  the  number  of 
roses  equals  the  number  of  pinks ;  how  many  has  she  of 
each  kind? 

5.  Divide  the  number  25  into  two  such  parts,  that  4 
times  one  part  shall  equal  the  other. 

6.  A  father  and  son  earned  in  one  week  $12;  how 
much  did  each  earn,  if  the  father  earned  twice  as  much 
as  the  son  ? 

7.  A  pole,  36  feet  in  length,  was  broken  into  twc 


MENTAL  ARITHMEIIC.  95 

unequal  p.eces,  such  that  |  of  the  longer  piece  equals 
the  shorter ;  required  the  length  of  each  piece. 

8.  In  a  certain  school,  consisting  of  35  scholars,  there 
were  |  as  many  girls  as  boys ;  how  many  boys  and  how 
many  girls  in  the  school  ? 

9.  The  sum  of  two  numbers  equals  40,  and  *  of  the 
greater  equals  the  less ;  required  the  numbers. 

10.  A  man  bought  a  horse  and  cow  for  $100,  and  the 
how  cost  |  as  much  as  the  horse ;  required  the  cost  of 
each. 

11.  Twice  the  sum  of  two  numbers  is  30,  and  3  times 
the  smaller  equals  twice  the  greater;  what  are  the  num- 
ners  ? 

12.  Two  thirds  of  the  number  of  dollars  that  A  and  B 
have  equals  40 ;  how  many  has  each,  if  5  times  A's  num- 
ber equals  7  times  B's  number  ? 

13.  Three  fourtks  of  40  is  |  of  the  number  of  apples 
and  pears  that  Reuben  has ;  how  many  has  he  of  each, 
if  3  times  the  number  of  apples  equals  7  times  the  num- 
ber of  pears  ? 

14.  Divide  36  apples  among  three  boys,  so  that  the 
second  may  have  twice  as  many  as  the  first,  and  the  third 
3  times  as  many  as  the  first. 

15.  Divide  66  plums   among  Ella,  Emma,  and  Ettie, 
so  that  Ella  shall  have  twice,  and  Emma  three  times  as 
many  as  Ettie. 

16.  A  watch  and  chain  cost  42  dollars ;  what  was  the 
cost  of  each,  provided  |  of  the  cost  of  the  watch  equals 
the  cost  of  the  chain? 

17  A,  B,  and  C,  together,  earned  $70;  A  earned 
twice  as  much  as  B,  and  B  twice  as  much  as  C ;  how 
much  did  each  earn  ? 

IS  The  sum  of  three  numbers  is  50  ;  the  second  is  3 
times  the  first,  and  the  third  is  twice  the  second ;  what 
are  the  numbers  ? 

19.  Harry  and  Thomas  lost  a  purse  of  money  contain- 
ing $24,  of  which  Harry  owned  ^  as  much  as  Thomas; 
how  mucli  did  each  lose  ? 


96  MENTAL  ARITHMETIC. 

20.  A  turkey,  duck,  and  hen,  cost  66  dimes,  the  duck 
cost  twice  as  much  as  the  hen,  and  the  turkey  4  times  as 
much  as  the  duck  ;  required  the  cost  of  each. 

21.  The  difference  between  two  numbers  is  27,  ami 
the  greater  is  4  times  the  smaller;  what  are  the  num- 
bers ? 

22.  Marie  has  40  cherries  more  than  Jane,  arid  5  times 
Jane's  number  equals  Marie's;  how  many  has  each? 

23.  Two  thirds  of  30  is  |-  of  the  difference  between  two 
numbers,  and  tho  less  is  |  of  the  greater;  what  are  the 
numbers  ? 

24.  A  man  bought  a  horse,  cow,  and  sheep  for  $105; 
how  much  did  he  pay  for  each,  provided  the  cow  cost  4 
times  as  much  as  the  sheep,  and  the  horse  4  times  as 
much  as  the  cow? 

25.  A  farmer  has   102   animals,  consisting  of  hogs, 
eheep,  and  cows ;  there  are  |  as  many  sheep  as  bogs,  and 
|  as  many  cows  as  hogs ;  required  the  number  of  each. 

26.  Of  a  certain  pole,  whose  parts  are  in  the  mud,  air, 
and  water,  |  of  the  length  in  the  air  equals  the  length 
in  the  water,  and  f  of  the  length  in  the  water  equals  the 
length  in  the  mud;  required  the  length  of  each  part,  sup- 
posing the  part  in  the  water  to  be  10  feet  longer  than  the 
part  in  the  mud. 


LESSON  V 

1 .  A  and  B  have  25  oranges ;  how  many  has  each,  if 
b  has  5  more  than  A? 

SOLUTION. — By  a  condition  of  the  problem,  A's  number  -j-  5 
oranges  equals  B's  number,  which,  added  to  A's,  is  twice  A'a 
number  -j-  5,  which  equals  25  oranges.  If  twice  A's  -|-  5  =  ?-5, 
twice  A's  =25  —  5,  or  20  ;  if  twice  A's  =  20,  once  A's  equals 
|  of  20,  which  is  10;  and  since  B  had  5  more  than  A?  10  -|-  £, 
or  Jo,  equals  B's  number.  Therefore,  &c. 


MENTAL  ARITHMETIC.  97 

2.  Mary  has  7  oranges  more  than  William,  and  they 
together  have  27 ;  how  many  has  each  ? 

3.  Stephen  has  10  cents  more  than  Martha,  and  they 
together  have  40  ;  how  many  has  each  ? 

4.  The  sum  of  two  numbers  is  31,  and  their  difference 
5  ;  what  are  the  numbers  ? 

5.  Thomas  and  Reuben  earned  the  same  sum  of  money, 
when  Reuben  found  $9,  and  they  then  together  had  $45; 
how  much  did  each  earn  ? 

6.  Ella  and  Kate  had  each  the  same  number  of  candies; 
Ella  eat  5  of  hers,  and  they  then  together  had  21 ;  how 
many  had  each  at  first  ? 

7.  Divide  the  number  28  into  two  such  parts,  that  one 
part  may  be  6  less  than  the  other. 

8.  Two  boys  found  an  equal  number  of  cents;  one  lost 
6,  and  the  other  4,  and  they  then  together  had  22 ;  how 
many  did  each  find? 

9.  A  and  B  had  equal  sums  of  money;  A  lost  $5,  and 
B  earned  $7,  and  they  then  together  had  $36 ;  how  much 
had  each  at  first? 

10.  Daniel  and  Edwin  had  each  the  same  number  of 
peaches;  Daniel  lost  6,  and  Edwin  gave  him  4,  and  they 
then  together  had  14 ;  how  many  had  each  then  ? 

11.  Three  times  Harry's  age,  increased  by  5  years, 
equals  Harvey's  age,  and  the  sum  of  their  ages  is  45 
years;  how  old  is  each? 

12.  Divide  the  number  48  into  two  such  parts,  that 
twice  the  first  part,  diminished  by  6,  shall   equal  the 
second  part  ? 

13.  The  sum  of  two  numbers  is  55,  and  the  greater 
equals  3  times  the  less,  diminished  by  5 ;  required  the 
numbers. 

14.  A  pole  whose  length  was  48  feet,  was  broken  Into 
two  unequal  pieces,  f  of  the  longer  part  equalling  the 
shorter  ;  required  the  length  of  each  piece  ? 

15.  A  tree  whose  length  was  45  feet  was  broken  into 
two  unequal  parts,  and  |  of  the  longer  piece,  plus  5  feet, 
equals  the  shorter     required  the  length  of  each  piece, 

9  * 


98  MENTAL  ARITHMETIC. 

16.  A  watch  and  chain  cost  $85,  and  T^  of  the  cost 
of  the  watch,  plus  .$7,  equals  the  cost  of  the  chain;  re- 
qui"ed  the  cost  of  each. 

17.  Francis  has  9  cents  more  than  ^  as  many  as  Fan 
nie,  and  they  together  have  42 ;   how  many  cents  has 
each  ? 

18.  A  cow  and  horse  cost  $132 ;  required  the  cost  of 
each,  if  the  cow  cost  |  as  much  as  the  horse,  minus  8 
dollars. 


19.  A  man  walked  110  miles  in  three  days;  he  walked 
5  miies  further  the  second  day  than  the  first,  and  10 
miles  further  the  third  than  the  second ;  how  far  did  he 
walk  each  day  ? 

20.  A  man  bought  a  sleigh,  horse,  and  harness,  for 
8152 ;  for  the  sleigh  he  gave  twice  as  much  as  for  the 
harness,  plus  $6,  and  for  the  horse  4  times  as  much  as 
for  the  harness,  plus  $6 ;  what  did  he  pay  for  each  ? 

21.  A  lady  bought  a  hat,  cloak,  and  shawl  for  $78; 
what  did  she  pay  for  each,  supposing  the  cloak  cost  twice 
as  much  as  the  hat,  plus  $4,  and  the  shawl  twice  as  much 
as  the  cloak,  lacking  4  dollars  ? 

22.  One  half  of  Mary's  oranges  equals  Annie's,  and  ^ 
of  Annie's  equals  Emma's,  and  they  together  have  28 ; 
how  many  has  each  ? 

23.  A  earned  f  as  much  as  B,  and  B  earned  |  as  much 
as  C,  and  they  together  earned  $108;  required  the  amount 
earned  by  each. 

24.  In  a  certain  field  there  are  42  animals,  consisting 
of  horses,  sheep,  and  cows;  required  the  number  of  each, 
provided  £  of  the  number  of  sheep,  -L  10,  equals  the 
number  of  cows,  and  J   of  the  number  of  sheep,  -j-  1(\ 
equals  the  number  of  horses. 


MENTAL  ARITHMETIC.  99 


SECTION    VI. 

LESSON  I. 

Percent.,  from  th  e  Latin  per,  by,  and  centum,  the  hundred,  mean? 
by  or  on  the  hundred.  Thus,  5  per  cent,  of  a  number  of  apples 
is  5  apples  of  a  hundred,  10  per  cent,  of  a  number  of  dollars  is 
10  dollars  on  a  hundred,  and  so  on,  whatever  be  the  denomina- 
tion. 

1.  At  a  gain  of  10  per  cent.,  what  part  of  the  value 
equals  the  gain  ? 

SOLUTION. — A  gain  of  10  per  cent,  is  a  gain  of  10  on  100. 
If  on  100  the  gain  is  10,  on  1  it  is     J^  of  10,  which  is  JH^,  or 
i  .    Therefore,  at  a  gain  of  10  per  cent,  J_  of  the  value  equals 
the  gain. 

2.  At  2,  4,  5,  or  8  per  cent.,  what  part  of  the  cost 
equals  the  gain  ? 

3.  At  a  loss  of  12,  14,  16,  or  20  per  cent.,  what  part 
of  the  value  equals  the  loss  1 

4.  If  I  gain  25,  30,  or  35  per  cent,  on  an  investment, 
what  part  of  the  money  invested  equals  the  gain  ? 

5.  A  gains  50  per  cent,  on  his  capital;  what  part  of 
the  capital  equalled  the  gain  ? 

6.  B  gained  at  one  time  60,  at  another  time  70,  and 
at  another  time  80  per  cent. ;  what  part  of  the  capital 
each  time  equalled  the  gain? 

7  What  part  of  the  cost  equals  the  gain  at  8-1,  12.\, 
16|.  or  33|  per  cent  ? 

8.  A  man  paid  SI 50  for  a  horse,  and  sold  it  at  a  gain 
of  30  per  cent.;  what  was  the  gain? 

SOLUTION. — At  a  gain  of  10  per  cent.  JjAc,  or  1  ,  of  the  cost 
equals  the  gain  1  of  $150  is  $15.  Therefore,  &c 


100  MENTAL  ARITHMETIC. 

9.  A  lady  bought  a  shawl  for  $8,  and  sold  it  at  a  gain 
of  25  per  cent. ;  required  the  gain. 

10.  A  merchant  sold  20  per  cent,  of  50  barrels  of 
flour ;  how  many  barrels  did  he  sell,  and  how  many  re- 
mained ? 

11.  Henry  sold  a  cow  worth  $40,  at  a  loss  of  5  per 
cent. ;  what  did  he  receive  for  the  cow  ? 

12.  Samuel  spent  20  per  cent,  of  $50  for  a  watch,  and 
20  per  cent,  of  the  remainder  for  a  chain ;  how  much  had 
he  remaining? 

13.  How  much  is  5  times  4  per  cent,  of  400  barrels  of 
flour,  and  6  times  5  per  cent,  of  300  barrels  ? 

14.  Which  is  the  greater,  and  how  much ;  20  per  cent, 
of  50  apples,  or  6  times  4  per  cent,  of  25  apples  ? 

15.  Thomas  having  a  horse  which  cost  $120,  sold  it  at  a 
gain  of  25  per  cent.,  and  the  buyer  sold  it  at  a  loss  of  20 
per  cent. ;  what  did  the  latter  receive  for  it  ? 

16.  A  lady  bought  6  yards  of  calico  for  180  cents,  and 
sold  it  at  a  gain  of  10  per  cent. ;  what  was  the  gain  on 
each  yard  ? 

17.  10  per  cent,  of  $300  dollars  is  f  of  what  Mary 
paid  for  a  shawl ;  required  the  cost  of  the  shawl. 

18.  A  merchant  purchased  10  barrels  of  flour  for  $50, 
and  sold  them  at  a  loss  of  20  per  cent. ;  what  did  ne  re- 
ceive for  each  barrel  ? 

19.  A  man  owning  T8ff  of  a  house,  sold  25  per  cent,  of 
it ;  how  much  had  he  remaining  ? 

20.  8  per  cent,  of  $200  is  f  of  what  A  gave  for  a 
watch ;  he  sold  it  so  as  to  gain  20  per  cent. ;  for  what  did 
he  sell  it  ? 

21.  A  and  B  together  have  $1600,  of  which  A  owns 
|  as  much  as  B;  A  then  obtains  20  per  cent,  of  B's  part  \ 
how  much  does  each  now  possess  ? 

22.  A  owned  50  acres  of  land,  and  B  owned  three 
times  as  much;  A  sold  B  20  per  cent,  of  his  land,  and 
fchen  bought  25  per  cent,  of  B's;  how  much  had  each 
after  this  operation? 


MENTAL  ARITHMETIC.  101 


LESSON  II. 

]  A  man  bought  a  watch  for  20  dollars,  and  sold  it 
for  $25 ;  what  was  the  gain  per  cent.  ? 

SOLUTION.— If  he  bought  it  for  $20,  and  sold  it  for  $25,  he 
gained  the  difference  between  $25  and  $20,  which  is  $5.  If 
on  $20  he  gained  $5,  on  $1  he  gained  ^  of  5,  which  is  ^,  or 
$1,  and  on  $100  he  would  gain  100  times  1,  which  are  l|o,  or 
$25;  hence  the  gain  is  25  per  cent. 

2.  A  boy  gave  25  cents  for  a  knife,  and  sold  it  for  30 
cents ;  what  did  he  gain  per  cent.  ? 

3.  A  lady  bought  a  shawl  for  $5,  and  sold  it  for  $8 ; 
what  was  the  gain  per  cent.  ? 

4.  Thompson  bought  a  boat  for  $20,  and  sold  it  for 
816 ;  what  was  the  loss  per  cent.  ? 

5.  Rose  bought  a  dress  for  $4,  and  sold  it  for  $6 ;  what 
was  the  gain  per  cent.  ? 

6.  Edwin  bought  a  horse  for  $150,  and  sold  it  for  |  of 
the  cost ;  required  the  loss  per  cent. 

7.  Robert  sold  his  horse  for  $150,  which  was  |  of 
what  he  paid  for  it ;  what  per  cent,  did  he  lose  ? 

8.  EHhu  bought  10  cows  for  $200,  and  sold  8  of  them 
for  what  they  all  cost ;  what  was  the  gain  per  cent.  ? 

9.  What  per  cent,  of  $25  is  $5  ?     Of  40  cows  is  8 
cows  ?     Of  60  apples  is  6  apples  ? 

10.  What  per  cent,  of  16  is  4  ?     Of  30  is  5  ?     Of  200 
is  8?     Of  80  is  4?     Of  96  is  12? 

11.  James  having  50  marbles,  sold  20  per  cent,  of 
them ;  what  per  cent,  of  the  whole  remained  ? 

12.  A  man  bought  25  barrels  of  flour ;   he  lost  20  per 
cent,  of  it,  and  sold  25  per  cent,  of  the  remainder;  what 
per  cent,  of  the  whole  remained  ? 

13.  What  per  cent,  of  \  is  £  ?     Of  ^  is  -^  ?     Of  \  is 
I?     Offis/,?     Of|isTy      Off1s|t 

14.  If  a  miller  takes  8  quarts  of  every  bushel  he  grinds 
fur  toll,  what  per  cent,  does  he  take  for  toll  1 


102  MENTAL  ARITHMETIC. 

15.  |  of  $6  is  twice  what  per  cent,  of  £  of  50  dollars  ? 

16.  4  of  $10  is  I  of  what  per  cent,  of  i  of  50  dollars  7 

17.  Two-thirds  of  90  dimes  is  f  of  what  Samuel  paid 
for  10  books;  he  sold  them  for  3  dimes  apiece;  required 
the  loss  per  cent. 

18.  A  merchant  bought  30  barrels  of  flour  for  $5  each, 
and  sold  f  of  them  at  the  rate  of  3  barrels  for  $24,  and 
the  rest  for  cost;  required  the  gain  per  cent. 

19.  ^  of  10  per  cent,  is  what  per  cent,  of  20  per  cent.  ? 

20.  |  of  8  per  cent,  is  what  per  cent,  of  30  per  cent.  ? 

21.  I  of  15  per  cent,  is  what  per  cent,  of  4^  times  16 
per  cent.  ? 

22.  Mary  sold  some  silk  for  $12,  and  thereby  cleared 
i  of  this  money ;  what  would  she  have  lost  per  cent,  if 
she  had  sold  it  for  6  dollars  ? 

23.  A  man  sold  a  cow  for  $25,  and  thereby  cleared  | 
of  this  mon<  y ;  how  much  would  he  have  gained  per  cent. 
if  he  had  sold  it  for  $30  ? 

24.  Willis  sold  some  books  for  $12,  and  thereby  cleared 
J  of  the  cost ;  what  would  he  have  lost  per  cent,  by  sell- 
ing them  for  $8  ? 


LESSON  III. 

1.  Thomas  sold  his  watch  for  $25,  and  thereby  gained 
25  per  cent. ;  what  was  the  cost  of  the  watch  ? 

SOLUTION. — Tf  he  gained  25  per  cent.,  then  ._2p_,  Cr  j,  of  the 
cost  equals  the  gain,  which  added  to  4,  the  cost,  is  «  of  the 
cost,  which  equals  $25.  If  |  of  the  cost  equals  $25,  |  of  the 
cost  equals  1  of  $25,  which  is  $5,  and  4,  or  the  cost,  equals  4 
times  $5,  which  are  $20.  Therefore,  &c. 

2.  Mary  sold  her  shawl  for  $14,  which  was  at  a  gain 
of  40  per  cent. ;  required  the  cost  of  the  shawl. 

3.  A  farmer  sold  a  cow  for  $23,  and  thereby  gained  15 
per  cent.  •,  required  the  value  of  the  cow. 


MENTAL  ARITHMETIC.  103 

4.  A  student  sold  his  library  for  $140,  and  thereby 
lost  30  per  cent. ;  what  was  its  value  ? 

5.  By  selling  a  hat  for  $8,  Mary  lost  20  per  ceot. ; 
what  was  the  value  of  the  hat  ? 

6.  A  dog  was  bought  for  §15,  and  sold  at  a  gain  of  20 
per  cent. ;  for  what  was  it  sold  ? 

7.  Mason  gained  20  per  cent,  by  selling  cloth  at  $6 
per  yard ;  how  should  he  have  sold  it  to  gain  25  per  cent.  ? 

8.  If  a  merchant  sells  muslin  at  39  cents  a  yard,  and 
thereby  gains  30  per  cent.,  how  ought  he  to  sell  it  to  lose 
40  per  cent.  ? 

9.  If  by  selling  land  at  $75  an  acre  I  gain  25  per  cent., 
how  must  I  sell  it  to  lose  40  per  cent.  ? 

10.  A  boat  was  sold  for  $91,  which  was  at  a  loss  of  85 
per  cent. ;  how  ought  it  to  have  been  sold  to  gain  40  per 
cent,  ? 

11.  Taylor  lost  60  per  cent,  on  a  watch,  by  selling  it  for 
840;  what  ought  he  to  have  received  to  gain  60  per  cent.  ? 

12.  Hinkston  sold  his  horse  and  carriage  for  $240,  and 
thereby  lost  4  per  cent. ;  what  would  he  have  gained  per 
cent,  by  selling  it  for  $300  ? 

1$.  A  wagon  was  sold  for  $90,  which  was  10  per  cent, 
less  than  its  value ;  what  would  have  been  the  gain  per 
cent,  if  it  had  been  sold  for  $120  ? 

14.  Mr.  Bowman  sold  2  books  for  $15  each ;  on  one  he 
gained  25  per  cent.,  and  on  the  other  he  lost  25  per  cent. ; 
how  much  did  he  lose  by  the  transaction  ? 

15.  A  tailor  sold  2   coats  for  $12  each;   on  one  h« 
gained  20  per  cent.,  and  on  the  other  he  lost  20  per 
cent. ;  did  he  gain  or  lose  by  the  sale,  and  how  much  ? 

16.  B  bought  a  watch  for  $42,  which  was  40  per  cent, 
le?s  than  its  value;  he  sold  it  for  30  per  cent,  more  then 
Us  value  ;  what  was  the  gain  ? 

1.7.  A  man  sold  2  watches  for  $80  each;  on  one  ho  lost 
20  per  cent.,  and  on  the  other  he  gained  25  per  cent.; 
how  much  was  gained  or  lost  by  the  transaction  ? 

18.  A  merchant  sold  a  stove  for  $30,  and  thereby  lost 
25  per  cent  ;  he  then  bought  another  for  $30,  and  upon 


104  MENTAL  ARITHMETIC. 

it  gained  25  per  cent. ;  what  was  gained  or  lost  by  the 
transaction  ? 

19.  Martha  sold  a  painting,  so  that  f  of  what  she  re 
eeived  for  it  equalled  |  of  the  cost ;  did  she  gain  or  lose, 
and  how  much  per  cent.  ? 

20.  Ten-el  sold  his  watch  and  chain  for  8120,  receiv- 
ing 5  times  as  much  for  the  watch  as  for  the  chain ;  on 
the  watch  he  gained  25  per  cent.,  and  on  the  chain  he 
lost  20  per  cent,  j  what  was  the  gain  ? 


LESSON  IV. 

1.  A  man  gained  25  per  cent,  by  selling  his  watch  for 
$20  more  than  it  cost ;  required  its  cost. 

2.  A  farmer  gained  80  per  cent,  by  selling  a  cow  for 
$9  more  than  she  cost ;  what  did  the  cow  cost  ? 

3.  A,  by  selling  his  dog  for  $6  less  than  it  cost,  lost 
15  per  cent,  on  the  sale  j  required  the  cost  of  the  dog. 

4;  A  hat  was  sold  for  20  cents  less  than  cost,  which 
was  at  a  loss  of  40  per  cent. ;  required  the  value  of  the 
hat. 

5.  Four  is  10  per  cent.,  5  is  20  per  cent.,  and  6  is  25 
per  cent.,  of  what  number? 

6.  Eight  is  40  per  cent.,  9  is  30  per  cent.,  and  12  is 
12  per  cent.,  of  what  number  ? 

7.  Thirty  is  25  per  cent,  less,  and  25  per  cent,  more, 
fchan  what  numbers  ? 

8.  A  man  gained  $20  by  selling  a  boat  for  20  per  cent 
more  than  its  value ;  what  would  he  have  gained  by  sell- 
ing  it  for  10  per  cent,  above  its  value  ? 

9.  A  piano  was  sold  for  $60  less  than  its  value,  which 
was  at  a  loss  of  30  per  cent. ;  what  would  have  been  the 
gain  per  cent,  if  it  had  been  sold  for  $250  ? 

10.  $24  is  4  per  cent,  of  the  sum  of  A's  and  B's  for- 
tune ;  how  much  money  has  each,  provided  A  has  twice 
as  much  as  B? 


MENTAL  ARITHMETIC.  105 

11.  An  agent  receives  $120  to  purchase  goods,  after 
deducting  his  commission,  which  is  20  per  cect.  on  the 
amount  expended;  required  his  commission. 

REMARK.—  ^,  his  commission,  -j-  4  what  he  expended,  =  | 
of  what  he  expended,  which  is  $120. 

12  A  man  receives  25  per  cent,  for  purchasing  goods, 
how  many  dollars  worth  can  he  purchase  with  §200,  re- 
taining his  commission  ? 

13.  A  receives  $216  to  buy  goods,  and  is  to  retain  8 
per  cent,  on  the  money  expended ;  required  the  amount 
of  money  expended. 

14.  Frick  received  $2800  to  invest  in  land,  after  de 
ducting  his  commission,  which  is  12  per  cent,  on  the 
amount  invested;  required  his  commission. 

15.  How  much  grain  must  a  farmer  take  to  mill  that 
he  may  bring  away  the  flour  of  2  bushels,  after  the  mil- 
ler has  taken  20  per  cent,  of  all  he  took  there  ? 

16.  A's  money  if  25  per  cent,  more  than  B's;  then  B's 
money  is  how  many  per  cent,  less  than  A's  ? 

17.  Morgan  sold  two  horses  for  $180  each  •  on  om  he 
gained  25  per  cent.,  and  on  the  other  he  lost  10  pez 
cent. ;  did  he  gain  or  lose,  and  how  much  ? 

18.  A's  shop  is  valued  at  $900 ;  for  what  sum  must  he 
have  it  insured,  at  10  per  cent.,  so  that  in  case  of  Joss  he 
may  receive  both  the  value  of  the  shop  and  premiu  ia  ? 

19.  At  5  per  cent.,  what  must  be  insured  on  a  house 
worth  $1900,  to  include  the  premium  in  case  of  loss  ? 

20.  At  8  per  cent.,  what  must  $2300  worth  of  pro- 
perty be  insured  for,  so  that  the  premium  may  be  included 
in  case  of  loss  ? 

21.  How  many  yards  of  cloth,  at  $4  a  yard,  must  a 
merchant  buy,  that  by  selling  it  at  a  profit  of  20  per  cent .. 
he  may  gain  $8  ? 

22.  A  man  receives  $530  to  purchase  sheep  and  cows; 
what  sum  will  he  expend  for  each,  after  deducting  his 
commission,  which  is  6  per  cent,  of  the*  money  expended, 
provided  he  expends  4  times  as  much  for  cows  as  sheep? 

10 


106  MENTAL  ARITHMETIC. 

LESSON  V 

INTEREST. 

Interest  is  money  charged  for  the  use  of  money  or  property. 
ft  is  estimated  at  a  certain  rate  per  cent.,  per  annum. 

The  Principal  is  the  sum  on  which  interest  is  computed. 

The  Amount  is  the  sum  of  the  principal  and  interest. 

The  Rate  per  cent,  is  the  interest  of  100  for  one  year. 

In  computing  interest  we  shall  consider  30  days  to  the  month, 
and  12  months  to  the  year. 

1.  Reduce  2  years  and  6  months  to  the  fraction  of  a 
year. 

SOLUTION. — In  1  yr.  there  are  12  months,  hence  1  month  is 
_i  of  a  year,  and  6  months  are  6  times  J^,  which  are  J^,  or  1 
of  a  year,  which,  added  to  2  years,  equals  2^,  or  5  years. 

2.  Reduce  each  of  the  following  to  the  fraction  of  a 
year :  2  yr.  8  mo.,  3  yr.  4  mo.,  4  yr.  3  mo.,  5  yr.  6  mo. 

3.  How  many  years  in  3  yr.  9  mo.  ?     7  yr.  2  mo.  ?     4 
yr.  10  mo.?     6  yr.  5  mo.?     6  yr.  8  mo.? 

4.  Reduce  3  years,  7  months,  15  days,  to  the  fraction 
of  a  year. 

SOLUTION. — There  are  30  days  in  a  month,  hence  1  day  is  J^, 
and  15  days  are  is  Or  1  of  a  month,  which,  added  to  7  mo., 
equals  71,  or  y>  months,  &c. 

5.  How  many  years  in  2  yr.  6  mo.  ?     2  yr.  8  mo  ?     3 
yr.  4  mo.  ?     4  yr.  3  mo.  ?     5  yr.  6  mo.  ? 

6.  How  many  years  in  3  yr.  9  mo.  ?     7  yr.  2  mo.  ?     4 
yr,  10  mo.?     6  yr.  5  mo.?     5  yr.  8  mo.? 

7.  How  many  years  in  2  yr.  2  mo  2  da  ?     3  yr.  3  mo. 
0  da  ?     2  yr.  4  mo.  5  da.  ? 

8.  How  many  years  in  4  yr.  7  mo.  6  da.  ?     5  yr.  5  mo. 
10  da.?     6yr.  2  mo.  12  da.9 

9.  How  many  years  in  7  yr.  3  mo.  18  da.  ?     8  yr.  6 
mo.  20  da.?     2  jr   1  mo.  6  da.? 


MENTAL  ARITHMETIC.  107 

10.  At  5  per  cent,  for  4  years,  what  part  of  the  princi- 
pal equals  the  interest  ? 

SOLUTION.— At  5  per  cent.  JgIJ  of  the  principal  equals  the 
iLterest  for  1  year,  and  for  4  yrs.,  4  times  Tg^,  which  are  ^g, 
or  i  of  the  principal  equals  the  interest. 

11.  At  10  per  cent,  for  5  years,  or  20  per  cent,  for  2 
years,  what  part  of  the  principal  equals  the  interest  ? 

12.  At  8  per  cent,  for  5  years,  or  9  per  cent,  for  10 
years,  what  part  of  the  principal  equals  the  interest? 

13.  At  7  per  cent,  for  5  years,  and  6  per  cent,  for  15 
years,  what  part  of  the  principal  equals  the  interest? 

14.  At  6  per  cent,  for  2  yr.  and  4  mo.,  what  part  of 
the  principal  equals  the  interest  ? 

15.  At  8  per  cent,  for  3  yr.  and  3  mo.,  what  part  of 
the  principal  equals  the  interest  ? 

16.  At  6.  per  cent,  for  5  yr.  and  8  mo.,  what  part  of 
the  principal  equals  the  interest  ? 

17.  At  7  per  cent,  for  12  yr.  and  6  mo.,  what  part  of 
the  principal  equals  the  interest  ? 

18.  At  8  per  cent,  for  1  yr.  4  rno.  15  da.,  what  part 
of  the  principal  equals  the  interest  ? 

19.  At  9  per  cent,  for  2  yr.  5  mo.  10  da.,  what  part  of 
the  principal  equals  the  interest  ? 

20.  At  5  per  cent,  for  3  yr.  7  mo.  6  da.,  what  part  of 
the  principal  equals  the  interest  ? 

21.  What  is  the  interest  of  $60  for  6  years,  at  5  per 
cent.  ? 

22.  What  is  the  interest  of  $40  for  4  years,  at  5  per 
cent.  ? 

23.  What  is  the  interest  of  $30  for  5  years,  at  4  per 
eent.  ? 

24.  What  is  the  interest  of  $80  for  7  years,  at  5  per 
cent.  ? 

25.  What  is  the  interest  of  $75  for  8  years,  at  6  per 
cent.  ? 

26.  What  is  the  interest  of  $50  for  9  years,  at  S  pel 
cent  ? 


108  MENTAL  ARITHMETIC. 

27.  What  is  the  interest  of  $250  for  6  years,  at  4  per 
cent.  ? 

28.  What  is  the  interest  of  $28  for  10  years,  at  5  per 
cent.  ? 

29.  What  is  the  interest  of  $400  for  11  years,  at  5  per 
cent.  ? 

30.  What  is  the  interest  of  $200  for  6^  years,  at  6  per 
oent.? 

31.  What  is  the  interest  of  $300  for  5|  years,  at  9  per 
cent.  ? 

32.  What  is  the  interest  of  $240  for  7^  years,  at  9  per 
cent.  ? 

What  is  the  interest 

33.  Of  $600  for  2  years,  3  months,  at  8  per  cent.  ? 

34.  Of  $300  for  4  years,  6  months,  at  6  per  cent.  ? 

35.  Of  $240  for  3  years,  9  months,  at  8  per  cent.  ? 

36.  Of  $225  for  6  years,  8  months,  at  6  per  cent.? 

37.  Of  $500  for  5  years,  4  months,  at  9  per  cent.  ? 

38.  Of  $330  for  7  years,  6  months,  at  4  per  cent.  ? 

39.  Of  $222  for  8  years,  4  months,  at  6  per  cent.  ? 

40.  Of  $666  for  6  years,  3  months,  at  8  per  cent.? 

41.  Of  $288  for  4  years,  2  months,  at  12  per  cent  ? 

42.  Of  $440  for  2  years,  1  month,  at  12  per  cent.? 

43.  Of  $120  for  5  years,  10  months,  at  12  per  cent  ? 

44.  Of  $540  for  3  years,  7  months,  6  days,  at  6  per 
cent.  ? 

45.  Of  $300  for  5  years,  3  months,  18  days,  at  10  per 
cent.  ? 

46.  Of  $400  for  2  years,  3  months,  9  days,  at  40  per 
cent.  ? 

47.  Of  $500  for  2  years,  2  months,  12  days,  at  5  per 
cont.?  ' 

48.  Of  $600  for  1  year,  6  months,  12  days,  at  15  per 
cent.  ? 

49.  Of  $200  for  1  year,  6  months,  20  days,  at  9  per 
cent.? 

50.  Of  $50C  for  3  years,  8  months,  12  days,  at  LO  per 
cent.  ? 


MENTAL  ARITHMETIC.  109 


LESSON  VI. 

1.  What  is  the  amount  of  $50  for  5  ytais,  at  3  per 
cent.  ? 

REMARK. — §  if  the  principal  equals  the  interest,  which  added 
to  4,  the  principal,  equals  1  of  the  principal,  which  equals  the 
amount.  |  of  $50  =  $70.° 

5 

2.  What  is  the  amount  of  $250  for  4  years,  at  5  pel 
cent.? 

3.  What  is  the  amount  of  $120  for  7  years,  at  10  per 
cent.  ? 

4.  What  is  the  amount  of  $400  for  5  years,  at  7  per 
cent.  ? 

5.  What  is  the  amount  of  $240  for  5  years,  at  5  per 
cent.  ? 

6.  What  is  the  amount  of  $200  for  2  years,  3  mo.,  at 
8  per  cent.  ? 

7.  What  is  the  amount  of  $600  for  7  years,  6  mo.,  at 
6  per  cent.  ? 

8.  What  is  the  amount  of  $300  for  3  years,  9  mo.,  at 

8  per  cent.  ? 

9.  What  is  the  amount  of  $320  for  8  years,  10  mo.,  at 
3  per  cent.  ? 

10.  What  is  the  amount  of  $400  for  7  years,  6  mo.,  at 
6  per  cent.  ? 

11.  What  is  the  amount  of  $360  for  8  years,  4  mo  ,  at 

9  per  cent.  ? 

12.  What  is  the  amount  of  $100  for  2  years,  6  months, 
20  days,  at  9  per  cent.  ? 

13.  A  and  B  wish  to  divide  the  amount  of  $500  for  8 
years,  at  5  per  cent.,  so  that  A's  part  shal)  be  6  times  B's ; 
required  the  share  of  each  ? 

14.  The  amount  of  $250  for  6  years,  at  10  per  cent., 
is  to  be  divided  between  C  and  D,  so  that  C  shall  have  % 
times  as  much  as  D :  what  does  each  receive  ? 

10* 


110  MENTAL  ARITHMETIC. 

15.  James  and  Henry  have  $1500  on  interest  for  4 
years,  at  10  per  cent. ;  what  amount  of  interest  will  each 
receive,  provided   James  has  twice  as  much  as  Henry  ? 

16.  A's  fortune  is  $200,  which  is  I  of-B's;   what 
interest  will  each  receive  on  his  money  in  4  years,  at  5 
per  cent.  ? 

17.  C's  money  is  $300,  which  is  |  of  D's ;  what  is  the 
amount  of  each  for  5  years,  at  6  per  cent.  ? 

18.  A's  money  is  $400,  which  is  f  of  B's;  how  much 
more  interest  will  B  receive  than  A,  in  8  years,  at  5  per 
cent.  ? 

19.  A,  B,  and  C,  together,  have  $1200,  of  which  A 
has  twice,  and  B  3  times  as  much  as  C;  what  is  the 
interest  of  each  for  5  years,  at  6  per  cent.  ? 

20.  If  the  interest  of  $2500  for  4  years,  at  10  per 
cent.,  be  divided  into  two  parts,  which  are  as  2  to  3,  it 
will  respectively  give  |  of  B's,  and  ^  of  A's  money;  how 
much  has  each? 


LESSON  VII. 

1.  What  principal  will,  in  6  years,  at  5  per  cent.,  give 
$60  interest? 

REMARK. — We  find  JL  Of  the  principal  equals  the  interest, 
which  is  $60.  If  _3^  of  the  principal  equals  $60,  -^  equals  J 
of  $60,  which  is  $20,  and  JO,  or  the  principal,  equals  10  times 
$20,  which  are  $200. 

2.  What  principal  will,  in  7  years,  at  5  per  cent.,  give 
$21  interest? 

3  What  principal  will,  in  8  years,  at  6  per  cent.,  give 
812  interest? 

4.  What  principal  will,  in  3  years,  at  8  per  cent.,  give 
$60  interest  ? 

5.  What  principal  will,  in  7  years,  at  4  per  cent.,  give 
$70  interest  ? 


MENTAL  ARITHMETIC.  Ill 

6.  What  principal  will,  in  8  years,  at  5  per  cent.,  give 
$60  interest  ? 

7.  What  principal  will,  in  3  years  and  4  months,  at  6 
per  cent.,  give  $80  interest? 

8.  What  principal  will,  in  7  years  and  6  months,  at  4 
per  cent.,  give  $90  interest? 

9    What  principal  will,  in  6  years  and  3  months,  at  8 
per  cent.,  give  $85  interest  ? 

10.  A  man  pays  $360  interest,  at  6  per  cent.,  annu- 
ally, on  money  borrowed;  what  is  the  sum  borrowed? 

11.  How  much  money  must  a  person  borrow,  that  he 
must  pay  an  annual  interest  of  $150,  at  5  per  cent.  ? 

12.  How  much  money  has  Howard  on  interest,  sup- 
posing he  receives  $320  for  5  years,  4  months,  at  6  per 
cent.  ? 

13.  The  interest  of  f  of  A's  money  for  6  years  and  3 
months,  at  4  per  coat.,  is  $250;  what  is  his  money? 

14.  The  interest  of  f  of  A's,  and  |  of  B's  fortune,  for 
5  years,  at  6  per  cent.,  is  $60,  and  $90,  respectively, 
required  the  fortune  of  each. 

15.  The  interest  of  the  sum  of  A's  and  B's  fortune, 
for  5  years,  at  7  per  cent.,  is  $210;  what  is  the  fortune 
of  each,  provided  B.  is  worth  twice  as  much  as  A? 

16.  Howard's  money  is  3  times  HowelPs,  and  in  5 
years,  at  8  per  cent.,  Howard  receives  $600  interest; 
how  much  money  has  each  ? 

17.  The  interest  on  i  of  A's  and  |  of  B's  fortune  for 
5  years,  at  6  per  cent.,  is  $240;  what  is  the  fortune  of 
each,  provided  ^  of  A's  equals  -|  of  B's? 

18.  The  interest  for  4  years,  at  5  per  cent.,  on  tie 
money  Martin  owes,  is  $40;   and  the  interest   for  the 
game  time  and  rate  per  cent.,  on  the  money  due  him, 
$70 ;  how  much  more  has  he  due  than  he  owes  ? 

19.  A's  money  is '4  times  B's,  and  the  sum  of  the 
interest  received  by  both  for  3  years,  at  8  per  cent,  is 
$600  ;  how  much  money  has  each  ? 

20.  The  interest  on  the  money  A  paid   for  a  farm, 
house,  and  store,  for  8  years,  at  5  per  cent,  is  equal  to 


112  MENTAL  ARITHMETIC. 

$18,000;  what  was  the  cost  of  each,  provided  the  farm 
cost  3  times  as  much  as  the  house,  and  the  house  twice 
as  much  as  the  store  ? 


LESSON  VIIL 

The  present  worth  of  a  debt,  payable  at  some  future  time, 
without  interest,  is  such  a  sum  as  would,  at  a  given  rate  pet 
cent.,  amount  to  the  debt  at  the  time  it  becomes  due.  The  pre- 
sent worth  may  therefore  be  found  in  the  same  manner  as  the 
principal,  when  we  have  given  the  amount,  time,  and  rate  per 
cent.  The  Discount  is  the  allowance  made  for  the  payment  of 
money  before  it  is  due.  It  equals  the  debt  minus  the  present 
worth. 

1.  What  principal  will,  in  8  yeaps,  at  5  per  cent., 
amount  to  $140  ? 

SOLUTION. — At  5  per  cent.,  _|>  of  the  principal  equals  the 
interest  for  1  year,  and  for  8  years,  8  times  — jj™  which  are  40 
or  2  of  the  principal  equals  the  interest,  which  added  to  £,  the 
principal,  equals  J  of  the  principal,  which  equals  the  amount, 
or  $140,  &c. 

2.  What   principal  will,  in  7  years,  at  6  per  cent., 
amount  to  $710  ? 

3.  What  principal  will,  in  4  years,  at  10  per  cent., 
amount  to  $420  ? 

4.  What  principal  will,  in  6  years  and  8  months,  at  9 
per  cent.,  amount  to  $820? 

5.  What  principal  will,  in  8  years  and  9  months,  at  8 
per  cent.,  amount  to  $510  ? 

6.  What  principal  will,  in  5  years  and  10  months,  at  6 
er  cent.,  amount  to  $540  ? 

7.  What  is  the  present  worth  of  $60,  due  4  years 
hence,  at  5  per  cent.  ? 

8  What  is  the  present  worth  of  $52,  due  5  years 
hence,  at  6  per  cent.  ? 


MENTAL  ARITHMETIC.  .      113 

9.  The  amount  of  |  of  B's  fortune,  for  3  years  and  4 
months,  at  6  per  cent.,  is  $600;  what  is  his  fortune? 

10.  The  amount  due  on  a  note  which  had  been  on 
interest  for  3  years  and  4  months,  at  9  per  cent.,  is  $520  j 
required  the  face  of  the  note. 

11.  What  is  the  present  worth  of  $270,  due  7  years 
hence,  at  5  per  cent.  ? 

12.  What  is  the  present  worth  of  $370,  due  8  years 
hence,  at  6  per  cent.  ? 

13.  What  is  the  discount  of  $580,  due  9  years  hence, 
at  5  per  cent.  ? 

14.  What  are  the  present  worth  and  discount  of  $340, 
due  10  years  hence,  at  7  per  cent.  ? 

15.  Required  the  discount  of  $700,  due  5  years  hence> 
at  8  per  cent. 

16.  Required  the  discount  of  $149,  due  7  years  hence, 
at  7  per  cent. 

17.  The  sum  of  A's  and  B's  money,  being  on  interest 
for  3  years  and  9  months,  at  8   per  cent.,  amounts  to 
$2600 ;  what  is  the  money  of  each  if  A's  is  3  times  B's  ? 

18.  A's  money  added  to  B's,  being  on  interest  for  5 
years  and  4  months,  at  6  per  cent.,  amounts  to  $660; 
what  sum  has  each  if  A's  is  4  times  B's  t 

19.  A  man  wishes  to  place  such  a  sum  of  money  on 
interest,  at  6  per  cent.,  that  it  will  give  an  annual  interest 
of  $360  for  a  poor  sister;  required  the  amount  invested. 

20.  Four  times  A's  money,  added  to  3  times  B's,  being 
on  interest  for  4  years,  at  10  per  cent.,  amounts  to  $4200 ; 
how  much  has  each,  if  3  times  B's  equals  A's? 

21.  Two  thirds  of  A's  fortune,  plus  |  of  B's,  being  oo 
interest  for  6  years,  at  5  per  cent.,  amounts  to  $7800 , 
what  is  the  fortune  of  each,  supposing  |  of  A's  equals  | 
ofBs? 

22.  |  of  the  cost  of  Bowman's  house,  plus  |  of  the  cost 
of  his  farm,  being  on  interest  for  5  years,  at  8  per  cent., 
amounts  to  $2100 ;  what  is  the  cost   of  each,  provided 
the  house  ^ost  ^  as  much  as  the  farm? 

23    Two  times  the  value  of  a  horse,  plus  8  timen  the 


114  MENTAL  ARITHMETIC. 

value  of  a  cow,  which  is  \  of  the  value  of  the  horse,  in  8 
years,  at  5  per  cent.,  gives  $84  interest \  required  the 
value  of  each, 

24.  The  money  Henry  paid  for  a  horse,  carriage,  and 
harness,  in  10  years,  at  5  per  cent.,  would  give  such  an 
interest,  that  it  on  interest  for  the  same  time,  and  rate, 
would  amount  to  $270 ;  how  much  did  he  pay  for  each, 
if  the  horse  cost  twice  as  much  as  the  carriage,  and  the 
carriage  3  times  as  much  as  the  harness  ? 


LESSON  IX. 

• 

1.  The  interest  of  $200,  for  a  certain  time,  at  5  pei 
cent.,  is  $60 ;  required  the  time. 

SOLUTION. — At  5  per  cent,  for  one  year,  -4*>  or  1  ,  of  the 
principal  equals  the  interest.  ^  of  $200  is  $10.  Iflt  require 
one  year  for  $200  to  gain  $10/"to  gain  $1  it  will  require  J^  of 
a  year,  and  to  gain  $60  it  will  require  60  times  1  of  a  year, 
which  are  £g  or  6  years.* 

2.  In  what  time  will  $100,  at  6  per  cent.,  give  $21 
interest  ? 

3.  In  what  time  will  $100,  at  7  per  cent.,  give  $14 
interest? 

4.  In  what  time  will  $200,  at  5  per  cent.,  give  $40 
interest  ? 

5.  In  what  time-  will  $150,  at  6  per  cent.,  give  $45 
interest? 

6.  In  what  time  will  $100,  at  8  per  cent ,  give  $32 
interest? 

7.  In  what  time  will  $300,  at  10  per  cent.,  give  $120 
interest? 

8.  In  what  time  will  $200,  at  8  per  cent.,  give  $48 
interest? 

*  The  latter  part  of  this  may  be  given  thus :  it  will  require  as 
many  years  as  $10  is  contained  times  in  $60,  which  are  6. 


MENTAL  ARITHMETIC,  115 

9.  In  what  time  will  $60,  at  5  per  cent.,  give  821 
interest? 

10.  In  what  time  will  $25,  at  6  per  cent.,  give  $9 
interest? 

11.  In  what  time  will  $50,  at  9  per  cent.,  give  $86 
interest? 

12.  In  what  time  will  $50,  at  4  per  cent.,  amount  to  $62  ? 

13.  In  what  time  will  $150,  at  5  per  cent.,  amount  to 
8210? 

14.  In  what  time  will  $300,  at  7  per  cent.,  amount  to 
$510? 

15.  In  what  time  will  a  principal  gain  twice  itself,  at  40 
per  cent.  ? 

REMARK. — 2  of  the  principal  =  the  interest  in  1  year,  and 
twice  the  principal  is  U)  ;  hence  it  will  require  as  many  years 
as  -  is  contained  times  in  U>. 

.->  o 

16.  In  what  time  will  a  principal  gain  3,  4,  and  5 
times  itself,  at  10  per  cent  ? 

17.  In  what  time  will  a  principal  double  itself,  at  5 
percent.?     At  6?  7?  8?  9?  10? 

18.  In  what  time  will  a  principal  double  itself;  at  10 
percent.?     At  12£?  15?  20?  25?  50? 

19.  in  what  time  will  a  principal  treble  itself,  at  5 
percent.?     At  10?  20?  25?  40?  50? 

20.  In  what  time  will  a  principal  quadruple  itself,  at 
5  per  cent.?     At  15?  30?  50?  60?  100? 

21.  The  amount  of  a  certain  principal,  for  a  certain 
time,  at  5  per  cent.,  is  $250,  and  the  amount  for  the 
same  time  at  8  per  cent,  is  $280 ;  required  the  principal 
and  time. 

22.  A  certain  sum  of  money,  on  interest,  amounts,  in 
a  certain  time,  at  6  per  cent.,  to  $310,  and,  at  10  per 
cent.,  for  the  same  time,  to  $350;  required  the  time  and 
principal. 


MENTAL  ARITHMETIC. 


LESSON  X. 

1.  At  what  per  cent,  will  $60,  in  5  years,  give  82] 
interest? 

SOLUTION. — For  5  years,  at  one  per  cent.,  _  £   ,  or    i_  of  the 
principal  equals  the  interest.       *    of  $60  is  $3.     If  $00  in  5 
years,  at  1  per  cent.,  gains  $3,  to  gain  $1,  it  will  require 
of  1  per  cent.,  and  to  gain  $21  it  will  require  21  times  JL,  whic! 
are  ^1,  or  7  per  cent.* 

2.  At  what  per  cent,  will  $40,  in  5  years,  give  $20 
interest  ? 

3.  At  what  per  cent,  will  $200,  in  3  years,  giv.e  $36 
interest  ? 

4.  At  what  per  cent,  will  $300,  in  4  years,  give  $00 
interest  ? 

5.  At  what  per  cent,  will  $80,  in  5  years,  give  $32 
interest  ? 

6.  At  what  per  cent,  will  $50,  in  6  years,  give  $15 
interest? 

7.  At  what  per  cent,  will  $60,  in  7  years,  give  $21 
interest  ? 

8  At  what  per  cent,  will  $50,  in  8  years,  give  $22 
interest  ? 

.   9.  At  what  per  cent,  will  $100,  in  4  years,  amount  to 
I]  20? 

10.  At  what  per  cent,  will  $90,  in  5  years,  amount  to 
$117? 

11.  At  what  per  cent,  will  $20,  in  7  yr.  6  mo.,  amount 
to  $26? 

12.  At  what  per  cent  will  $6,  in  3  yr.  4  mo.,  amount 
to  $7? 

*  The  latter  part  of  this  may  be  given  thus: — It  will  require 
as  many  times  1  per  cent,  as  $3  is  contained  times  in  $21, 
which  are  7. 


MENTAL  ARITHMETIC.  117 

13.  At  what  per  cent,  will  a  given  principal  gain  3 
times  itself  in  10  years  ? 

REMARK. — JL  of  the  principal  equals  the  interest  at  1  per 
cent.  ;  hence  it  will  require  as  many  times  1  per  cent,  as  1  is 
contained  times  in  |g. 

14.  At  what  per  cent,  will  a  principal  gain  2,  4,  5, 
and  6  times  itself  in  30  years  ? 

15.  At  what  per  cent,  will  a  principal  double  itself  in 
4  years?     In  10?  12?  20? 

16.  At  what  per  cent,  will  a  principal  double  itself  in 
8  years?     In  25?  331?  50? 

17.  At  what  per  cent,  will  a  principal  treble  itself  in 
10  years?     In  20?  25?  40?  80?  100? 

18.  At  what  per  cent,  will  a  principal  quadruple  itself 
in  10  years?     In  15?  30?  60?  100?  150? 

19.  At  what  per  cent,  will  a  principal  quintuple  itself 
in  4  years?     In  20?  40?  80?  100?  200? 

20.  A  gained  $20  by  selling  an  article  for  20  per  cent, 
more  than  cost }  required  the  cost  and  amount  received 
for  it. 

21.  B  lost  25  per  cent,  by  selling  a  boat  for  20  per 
cent,  of  $150 ;  required  the  value  of  the  boat  and  the 
amount  received  for  it. 

22.  A  man  sold  2  cows  for  $50,  gaining  25  per  cent, 
on  the  first,  and  losing  25  per  cent,  on  the  second;  what 
was  the  value  of  each,  if  he  received  |  as  much  for  the 
second  as  for  the  first  ? 

23.  The  amount  of  a  certain  principal  for  7  years,  at 
a  certain  per  cent.,  is  $540,  and  for  10  years,  $600;  re- 
quired the  principal  and  rate  per  cent. 

24.  The  amount  of  a  certain  principal  for  4  years,  at 
a  certain  per  cent.,  is  $420,  and  for  9  years,  at  the  same 
rate,  $570 ;  required  the  rate  per  cent,  and  principal. 


118  MENTAL  ARITHMETIC. 


SECTION    VII. 

LESSON  I. 

1.  &  *»<?  B  hired  a  pasture  for  $36.     A  pastured  4 
and  B  5  cows ;  how  much  should  each  pay  ? 

SOLUTION. — If  A  pastured  4  cows  and  B  5,  they  both  pastured 

4  +  6,  which  are  9  cows.     If  the  pasturage  of  9  cows  cost  $36, 
the  pasturage  of  1  cow  will  cost  ^  of  $36,  which  is  $4.  and 
the  pasturage  of  4  cows,  A's  number,  will  cost  4  times  $4,  or 
$16,  and  the  pasturage  of  5  cows,  B's  number  will  cost  5  times 
$4,  or  $20. 

2.  Two  boys  bought  60  apples  for  12  cents;  one  paid 

5  cents,  and  the  other  7  cents ;  how  many  apples  should 
each  receive  ? 

3.  Rufus  and  William  paid  20  cents  for  40  peaches, 
of  which  Rufus 'paid  9,  and  William,  11  cents;  how  many 
peaches  belong  to  each  ? 

4.  Three  men  hired  a  horse  for  20  days,  at  the  rate  of 
$1  per  day;  the  first  used  it  5,  the  second,  6,  and  the  third, 

9  days ;  how  much  should  each  pay  ? 

5.  A  and  B  hired  a  pasture  for  $44;  A  puts  in  12 
oxen,  and  B  100  sheep;    how  much  should  each  pay, 
supposing  an  ox  to  eat  as  much  as  10  sheep  ? 

6.  Two  farmers  hire  a  pasture  for  $56 ;  one  turns  in 

10  cows,  and  the  other  36  horses;    how  much  should 
each  pay,  provided  a  cow  eats  twice  as  much  as  a  horse? 

7.  Three  men,  A,  B,  and  C,  bought  144  bushels  of 
peaches  for  $72,  of  which  A  paid  J,  B,  f ,  and  C,  the  re ' 
mainder;  how  many  bushels  did  each  receive? 

8.  A  and  B  engage  to  do  a  piece  of  work  for  $72 ;  A 
sends  6  men,  and  B,  15  boys;  how  much  should  each 
receive,  supposing  2  men  to  do  as  much  as  3  boys  ? 

9    A  and  B  agree  to  mow  a  field  of  grass  for  $54 ;  A 


MENTAL  ARITHMETIC.  119 

sends  3  men  5  days,  and  B  sends  4  n  en  3  days ;  how 
much  should  A  and  B  receive  respectively  ? 

10.  Two  men  hire  a  lot  of  pasture  for  $10 ;  one  turns 
in  6  horses  for  7  days,  and  the  other,  7  horses  for  4 
days  ;  how  much  should  each  pay  ? 

11.  A  and  B  built  a  boat  for  $140;  A  sent  6  men  5 
days,  and  B,  4  men  10  days;  how  much  should  A  and  B 
receive  respectively? 

12.  Two  men  gain  in  trade  $440;  A  put  in  $25  for  4 
nnonths,  and  B,  $15  for  8  months;  what  is  each  man's 

share  of  the  gain  ? 

13.  C  and  D  build  a  wall  for  $120 ;  C  with  4  assistants 
laboured  4  days,  and  D  with  3  assistants  laboured  5  days ; 
how  much  do  C  and  D  receive  respectively  ? 

14.  A,  B,  and  C  build  a  boat  for  $80 ;  A  sent  3  men 
4  days.  B  5  men  2  days,  C  3  men  6  days ;  how  much  do 
A,  B,  and  C  receive  respectively  ? 

15.  A  and  B  plough  a  field  for  $76 ;  A  employed  12 
horses,  and  B  18  oxen ;  they  completed  it  in  4  days ; 
what  was  the  value  of  the  daily  labour  of  each  horse  and 
ox,  supposing  3  horses  do  as  much  as  5  oxen  ? 

16.  E  and  F  engaged  to  reap  a  field  of  wheat  for  $54 ; 
E  sent  3  men  5  days,  and  F,  6  boys  4  days;  how  much 
should  each  receive,  if  1  man  does  as  much  as  2  boys  ? 

17.  In  a  field  of  grass,  which  cost  $24,  M  turned  16 
horses  for  3  weeks,  and  N,  25  cows  for  4  weeks;  how 
much  should  each  pay,  if  4  horses  eat  as  much  as  5  cows? 

18.  R,  S,  and  T  hire  a  pasture  for  $63 ;  R  puts  in  6 
horses,  S  puts  in  18  cows,  and  T,  48  sheep ;  how  mucL 
should  each  pay,  if  a  horse  eat  twice  as  much  as  a  cow, 
and  a  cow  4  times  as  much  as  a  sheep  ? 


120  MENTAL  ARITHMETIC. 


LESSON  II. 

1 .  Divide  30  cents  between  A  and  B;  so  that  their 
shares  will  be  to  each  other  as  4  to  6. 

SOLUTION. — Since  the  shares  are  to  be  to  each  other  as  4  to  6,  if 
we  divide  30  cents  into  4  -f  6,  which  aro  10  equal  parts,  4  uf 
these  parts,  or  j^,  will  be  A's,  and  6  of  these  parts,  or  ^  will 
be  B's  number,  &c. 

2.  Divide  45  apples  between  Thomas  and  Harry,  so 
that  their  shares  may  be  to  each  other  as  3  to  2. 

3.  Divide  the  number  50  into  two  parts,  that  shall  be 
to  each  other  as  7  to  3. 

4.  In  a  school  consisting  of  45  pupils,  there  are  5  girls 
for  every  4  boys ;  how  many  of  each  sex  in  the  school  ? 

5.  The  sum  of  two  numbers  is  40,  and  the  larger  is  to 
the  smaller  as  5  to  3 ;  required  the  numbers. 

6.  Divide  45  plums  among  three  boys,  so  that  their 
shares  may  be  in  the  proportion  of  2,  8,  and  4. 

7.  Two  men  bought  a  barrel  of  flour  for  $8,  the  first 
paying  $3,  and  the  second  $5;  how  .much  of  the  flour 
belongs  to  each  ? 

8.  Three  men  bought  75  horses,  and  as  often  as  the 
first  paid  $4,  the  second  paid  $5,  and  the  third,  $6 ;  how 
many  horses  should  each  receive  ? 

9.  Divide  $44  between  A  and  B,  so  that  B  shall  have 
$3^  as  often  as  A  $2. 

10.  The  sum  of  2  numbers  is  50,  and  the  first  is  to  the 
second  as  4  to  | ;  what  are  the  numbers  ? 

11.  Divide  the  number  49  into  two  parts  which  are  to 
each  other  as  |  to  |. 

12.  The  sum  of  three  numbers  is  46;  what  is  eacb 
of  the  numbers,  if  they  are  to  each  other  as  ^,  f ,  and  f  ? 

13.  Divide  the  number  50  into  3  parts  which  shall  be 
o  each  other  as  |,  1£,  and  2. 

14.  A,  B,  and  C  found  a  purse  containing  $1 00,  which 


MENTAL  ARITHMETIC.  121 

fchey  agree  to  divide  in  the  proportion  of  |,  |,  and  1£ ; 
how  much  does  each  receive  ? 

15.  A  and  B  agree  to  pay  $25  toward  building  a 
church,  which  is  to  be  situated  2  miles  from  A,  and  8 
miles  from  B;  how  much  does  each  contribute,  if  the} 
pay  in  proportion  to  the  reciprocals  of  their  distances  ? 
,  16.  If  $420  be  divided  into  two  parts,  which  are  to 
each  other  as  4  to  |,  it  will  respectively  give  |  of  A's, 
and  |  of  B's  fortune ;  required  the  fortune  of  each. 

17.  If  the  interest  of  $500  for  4  years,  at  5  per  cent., 
be  divided  into  two  parts,  to  each  other  as  2  to  3,  it  will 
respectively  give  f  of  A's,  and  |  of  B's  fortune;  required 
the  fortune  of  each. 


18.  A's  fortune  added  to  ^  of  B's,  being  on  interest 
for  5  years  at  6  per  cent.,  equals  $2600;  what  is  the  for- 
tune of  each,  provided    A's  is  to  B's  as  3  to  4  ? 

19.  |  of  A's  fortune,  plus  \  of  B's  fortune,  being  on 
interest  for  6  years  at  10  per  cent.,  amounts  to  $800: 
what  is  the  fortune  of  each,  if  A's  fortune  is  to  B's  as  9 
to  8? 

20.  M's  fortune,  + 1  of  N's,  which  is  equal  to  £  of 
M's,  is  $900,  and  if  the  sum  of  M's  and  N's  be  divfded 
in  the  proportion  of  ^  to  f ,  it  will  respectively  give  \  of 
K's  and  |  of  T's  fortune ;  required  the  fortunes  of  each. 


LESSON  III, 

1.  A  can  do  a  piece  of  work  in  4  days ;  what  part  of 
it  can  he  do  in  one  day  ? 

2.  B  can  cut  a  cord  of  wood  in  |  of  a  day;  how  much 
can  he  cut  in  one  day  ? 

3.  A  man  can  build  |  of  a  boat  in  a  week;  how  long 
will  it  require  for  him  to  build  the  whole  boat? 

11* 


122  MENTAL  ARITHMETIC 

4.  A  mason  can  build  a  wall  in  2i  days;  what  part  of 
it  can  he  build  in  one  day  ? 

5.  If  A  and  B  can  mow  |  of  a  field  of  grass  in  one  day. 
how  long  will  it  require  to  mow  the  whole  field  ? 

6.  A  can  do  a  piece  of  work  in  3  days,  and  B  in  6 
days ;  what  part  can  each  do  in  one  day  ? 

7.  If  B  can  do  a  piece  of  work  in  3  days,  and  C  in  6 
days,  how  much  can  they  together  do  in  one  day  ? 

8.  If  B  and  C  can  do  |  of  a  piece  of  work  in  one  day, 
how  long  will  it  require  to  do  the  whole  work  ? 

9.  Fuller  can  eat  a  bushel  of  apples  in  4  days,  and 
Brodhead  in  6  days ;  how  many  days  would  it  last  them 
both? 

10.  A  can  dig  a  ditch  in  5  days,  and  B  in  6  days;  in 
what  time  will  they  do  it  working  together? 

11.  C  can  make  a  chest  in  4  days,  and  D  in  7  days; 
in  what  time  can  they  make  it  working  together  ? 

12.  E  can  reap  a  field  in  6  days,  and  F  in  8  days;  how 
long  will  it  take  them  both  to  reap  it? 

13.  A  can  do  a  piece  of  work  in  3  days,  B  in  4  days, 
and  G  in  6  days;  in  what  time  can  they  together  do  it? 

14.  A  cistern  has  two  pipes,  by  the  first  of  which  it 
may  be  filled  in  12  hours,  and  by  the  second,  in  |  of  the 
time;  how  long  will  both  be  in  filling  it? 

15.  A  can  make  a  book-case  in  6  days,  and  A  and  B 
can  make  it  in  4  days;    in  what  time  can  B  make  it 
alone  ? 

16.  A,  B,  and  C  can  dig  a  ditch  in  3  days.     A  can 
dig  it  in  6  days,  and  B  in  8  days ;  in  what  time  can  0 
alone  dig  it  ? 

17.  A  pound  of  tea  lasted  a  man  and  wife  3  months, 
and  the  wife  alone,  4  months ;  how  long  would  it  last  the 
man  alone  ? 

18.  A,  B,  and  C  can  mow  a  field  in  4  days,  A  and  B 
IB  6  days,  and  B  and  C  in  9  days ;  how  long  will  it  take 
each  to  mow  it  ? 

19.  A,  B,  and  C  can  dig  a  ditch  in  6  days,  A  and  B 


MENTAL  ARITHMETIC.  123 

in  8  days,  and  B  in  12  days ;  how  long  will  it  take  each 
to  do  it  ? 

20.  If  3  men,  or  4  boys,  can  do  a  piece  of  work  in  12 
days,  in  what  time  can  3  men  and  4  boys  do  it  ? 

21.  If  A  can  do  a  piece  of  work  in  |  of  a  day,  and  B 
in  |  of  a  day,  how  long  will  it  take  both  to  do  it  ? 

22.  C  can  cut  a  cord  of  wood  in  |  of  a  day,  and  D  tc 
|  of  a  day ;  in  what  time  can  they  together  cut  a  cord  ? 

23.  D  can  make  a  fence  in  9  days,  and  I)  and  E  in  6 
days ;  how  long  will  it  take  E  to  make  what  remains  after 
D*has  built     of  it? 


24.  Two  men,  or  3  boys,  can  plough  an  acre  in  |  of  a 
day ;  how  long  will  it  require  3  men  and  2  boys  to  do  it? 

25.  A  can  plough  a  field  in  J  of  a  day,  B  in  |  of  a  day, 
and  C  in  |  of  a  day ;  how  long  will  it  take  them  togethei 
to  plough  the  field  ? 

26.  A,  B,  and  C  can  mow  a  field  in  6  days,  and  A  and 
B  in  9  days  •  after  the  three  had  worked  2  days,  C  left ; 
how  long  did  it  require  A  and  B  to  finish  it  ? 

27.  Marie  can  make  a  dress  in  6  days,  Sallie  in  ^  of 
the  time,  and  Ewretta  in  |  of  the  time ;  in  what  time  can 
Marie  and  Sallie  finish  it,  after  the  three  have  worked  | 
of  a  day  ? 

28.  A  can  build  a  boat  in  |  of  a  month,  and  B  in  |  of 
a  month ;  after  A  had  wrought  |  of  a  month  B  joined 
him ;  how  long  was  the  boat  building  ? 

29.  Amos  can  plough  25  per  cent,  of  a  field  in  a  month, 
and  Anson,  45  per  cent. ;  after  they  both  had  worked  2 
weeks,  how  long  would  it  require  Amos  to  finish  it? 

30.  A,  B,  and  C  can  build  a  vessel  in  \  of  a  year,  A 
and  C  in  ^  of  a  year,  and  C  in  |  of  a  year;  after  they  had 
all  laboured  1  month,  A  left ;  in  what  time  could  B  and 
C  finish  it  ? 


1 24  MENTAL  ARITHMETIC . 


LESSON  IV. 

1,  A  and  B  together  have  34  apples,  and  f  of 
number  equals  |  of  B's  number;  how  many  has  eacl 


to  I  of  B's,  equals  \i  of  B's,  which  equals  34  apples,  &c. 

2.  Thomas  and  Walton  together  have  $55,  and  f  of 
Thomas's  money  equals  |  of  Walton's ;  how  much  haa 
each? 

3.  The  sum  of  two  numbers  is  28,  and  |  of  the  smaller 
equals  \  of  the  greater ;  what  are  the  numbers  ? 

4.  Divide  46  oranges  between  Chester  and  Henry,  so 
that  |  of  Chester's  may  equal  f  of  Henry's  number. 

5.  4  of  the  number  of  apple-trees  in  an  orchard  equals 
^  of  tne  number  of  peach-trees,  and  in  all  there  are  60 
trees ;  required  the  number  of  each. 

6.  A  pole,  whose  length  was  63  feet,  was  broken  into 
two  parts,  such  that  f  of  -the  first  part  equals  |  of  the 
second ;  required  the  length  of  each  piece. 

7.  The  sum  of  two  numbers  is  69 ;  what  is  each  of 
the  numbers,  provided  they  are  to  each  other  as  f  to  |  ? 

8.  Walter  bought  a  hat  and  coat  for  $26,  and  2^  times 
the  cost  of  the  hat  equals  f  of  the  cost  of  the  coat;  required 
the  cost  of  each. 

9.  Says  B  to  C,  f  of  my  age,  +  6  years  equals  |  of 
yours,  and  the  sum  of  our  ages  is  42  years ;  required  the 
ago  of  each. 

10.  The  difference  between  two  numbers  is  6,  and  | 
of  the  first  equals  |  of  the  second ;  what  are  the  num- 
bers? 

11.  Fanny  has  14  plums  more  than  Sa Jie,  and  |  of 
Fanny's    equal?  |  of   Sallie's   number;    how  many  has 
each? 


MENTAL  ARITHMETIC.  125 

12.  £  of  the  difference  between  two  numbers  is  6,  and 

;of  the  first  number  equals  \  of  the  second;  required 
e  numbers. 

13.  |  of  A's  money  equals  $32,  and  |  of  B's  money 
equals  |  of  A's;  how  much  has  each  ? 

14.  Five  sixths  of  the  difference  between  A's  and  B's 
fortune  is  $500,  and  f  of  A's  equals  |  of  B's  fortune ; 
what  is  the  fortune  of  each  ? 

15.  A  and  B  can  build  J  of  a  boat  in  a  day,  and  twice 
what  A  builds  equals  what  B  builds;  how  much  can  each 
build  in  one  day  ? 

16.  Two  boys  can  do  a  piece  of  work  in  6  days,  and 
twice  what  A  does  equals  what  B  does ;  how  long  will  it 
take  each  to  do  it  ? 

17.  Two  pipes  fill  a  cistern  in  15  hours,  and  |  of  what 
one  pours  in  equals  §  of  what  the  other  pours  in ;  how 
long  will  it  take  each  to  fill  it  ? 


18.  John  and  Henry  can  mow  60  acres  of  grass  in  6 
weeks,  and  4  of  what  John  can  mow  in  a  day  equals  what 
Henry  can  mow  in  a  day;  how  long  will  v  it  take  each  to 
mow  it  ? 

19.  Three  men  can  build  a  boat  in  6  days,  and  the 
parts  they  each  build  are  to  each  other  as  1,  2,  and  3 ;  in 
what  time  co  ild  each  build  it  alone? 

20.  A  horse  and  cow  eat  a  quantity  of  hay  in  3  weeks; 
now  long  will  it  last  each,  provided  the  horse  eat  only  | 
as  much  as  the  cow  ? 

21.  Three  men,  A,  B,  and  C,  can  build  a  boat  in  12 
days,  and  their  rates  of  working  are  as  4,  -|,  and  |;  how 
long  would  it  take  each  alone  to  build  it  ? 

22.  B  can  drink  a  keg  of  mead  in  4  days;  f  of  what 
B  drinks  equals  i  of  what  A  drinks,  and  \  of  what  0 
drinks.     After  the  three  had  been  drinking  -f.  of  a  day, 
A  and   C  drink  the  remainder;   how  long  did  it  take 
•iheni  ? 


MENTAL  AR1TMMETJO. 


LESSOR  V. 

1.  A  mat  receives  $3  a  day  for  his  labour,  and  forfeits 
$1  each  day  he  is  idle,  and  at  the  expiration  of  30  days 
receives  $50 ;  how  many  days  was  he  idle  ? 

SOLUTION. — Had  he  laboured  30  days,  he  would  have  received 
30  times  $3,  or  $90;  he  therefore  lost  $90  —  $50,  which  is  $40, 
by  his  idleness.  Every  day  he  was  idle  he  lost  $3,  his  wages, 
plus  $1,  his  forfeit,  which  are  $4.  If  in  one  day  he  lose  $4,  to 
lose  $1  it  will  require  ^  of  a  day,  and  to  lose  $40,  40  times  J, 
which  are  10  days,  the  number  of  days  he  was  idle. 

2.  Warner  receives  $2.50  a  day  for  his  labour,  and 
pays  50  cents  for  his  board,  and  at  the  expiration  of  40 
days  he  saves  $50 ;  how  many  days  was  he  idle  ? 

3.  A  labourer  agreed  to  work,  on  the  condition,  that 
for  every  day  he  worked  he  should  receive  $2,  and  for 
every  day  he  was  idle  he  should  forfeit  50  cents ;  how 
many  days  did  he  labour,  if  at  the  end  of  25  days  he  re- 
ceived $30? 

4.  A  boy  agreed  to  carry  32  glasses  to  a  certain  store 
for  5  cents  apiece,  on  condition  that  for  each  one  he  broke 
he  should  forfeit  10  cents;  he  received  $1.     How  many 
did  he  break  ? 

5.  James  engaged  to  labour  on  condition  that  for  every 
day  he  worked  he  should  receive  $1^,  and  for  every  day 
he  played  he  should  pay  $^  for  his  board ;  at  the  expira- 
tion of  30  days  he  received  $35.     How  many  days  did  be 
work? 

6.  Eobeit  agreed  to  carry  50  oranges  to  market  for  £ 
a  cent  each,  on  condition  that  he  should  forfeit  2^  centa 
for  each  one  he  eat ;  he  received  16  cents.     How  many 
did  he  eat  ? 

7.  The  head  of  a  fish  is  10  inches  long,  the  tail  is  as 
long  as  the  head,  plus  '£  of  the  body,  and  the  body  is  as 


MENTAL  ARITHMETIC.  127 

long  as  the  head  and  tail  both ;  required  the  length  of 
the  fish, 

SOLUTION. — Since  the  tail  is  as  long  as  the  head,  +  ^  °f  ^e 
body,  i  of  the  length  of  the  body,  -j-  10  inches,  equals  the  length 
of  the  tail,  "which,  added  to  the  length  of  the  head,  equals  £  of 
the  length  of  the  body,  -|-  20  inches,  which,  by  the  condition  of 
the  problem,  equals  |  of  the  length  of  the  body.  If  |  of  the 
length  of  the  body  equals  1  of  the  length  of  the  body,  -f  20, 
2  —  ^,  or  £  of  the  length  of  the  body,  must  equal  20  inches,  &c. 

8.  The  head  of  a  fish  is  8  inches  long,  the  tail  is  as 
long  as  the  head,  plus  |  of  the  body,  and  the  body  is  as 
long  as  the  head  and  tail  both ;  required  the  length  of  the 
fish. 

9.  The  head  of  a  perch  is  4  inches  long,  the  tail  is  as 
long  as  the  head,  plus  ^  of  the  body,  and  the  body  is  as 
long  as  the  head  and  tail  both;  what  is  the  length  of  the 
perch  ? 

10.  A  has  6  cents,  B  has  as  many  as  A,  plus  |  as  many 
as  C,  and  C  has  as  many  as  A  and  B  both ;  required  the 
number  of  cents  possessed  by  each  and  all. 

11.  The  tail  of  a  pike  weighs  3   ounces,  the  head 
weighs  as  much  as  the  tail,  plus  \  of  the  weight  of  the 
body,  and  the  body,  twice  as  much  as  the  head  and  tail; 
required  the  weight  of  the  fish. 

12.  A  tree  by  falling  was  broken  into  3  pieces;  trie 
bottom  piece  was  5  feet  long,  the  top  as  long  as  the 
bottom,  plus  4-  of  the  middle,  and  the  middle  3  times  as 
long  as  the  other  two  pieces ;  what  was  the  length  of  each 
piece,  and  the  tree  ? 

13.  A  agreed  to  labour  a  certain  time  for  $60,  on  con- 
dition, however,  that  for  each  day  he  was  idle  he  should 
forfeit  $2  ;  at  the  expiration  of  the  time  he  received  $30  : 
how  many  days  did  he  labour,  supposing  he  receives  $2 
a  day  for  his  labour  ? 

14.  The  head  of  a  trout  weighs  2  pounds,  the  tai! 
weighs  2  poun  is  more  than  the  head,  pJ  as  \  of  the  body, 


128  MENTAL  ARITHMETIC. 

and  the  body  weighs  as  much  as  the  head  and  tail  to- 
gether ;  required  the  weight  of  the  fish. 

15.  A  man  left  $26,000  to  his  wife,  son,  and  daughter, 
on  condition  that  if  the  daughter  died  before  becoming 
of  age  the  widow  should  have  |  of  the  fortune,  but  if  the 
son  died  the  widow  should  have  |  of  it;  required  the 
share  of  each  if  they  all  live. 

REMARK. — It  will  be  readily  seen  that  the  widow  has  three 
times  as  much  as  the  daughter,  and  the  son  three  times  as  much 
as  the  widow,  which  is  9  times  as  much  as  the  daughter. 

16.  A  man  wishing  to  erect  some  buildings,  concluded 
that  if  he  built  a  store  and  house,  the  store  should  cost  J 
of  his  money,  but  if  he  built  a  house  and  barn,  the  barn 
should  cost  J  of  his  money ;  what  was  the  cost  of  each, 
supposing  he  built  all  three,  and  his  fortune  was  $77,000? 

17.  A  gentleman  receives  $4  per  day  for  his  labour,  and 
pays  $8  per  week  for  his  board ;  at  the  expiration  of  1 0 
weeks  he  has  saved  $144 ;  required  the  number  of  idle 
and  working  days. 

18.  A  man  having  a  daughter  in  France,  and  a  son  in 
Spain,  willed,  if  the  daughter  returned,  and  not  the  son, 
the  wife  should  have  |  of  the  fortune,  but  if  the  son  re- 
turned, and  not  the  daughter,  he  should  have  f  of  the 
fortune ;  they  both  returned,  and  it  was  found  that  the 
son  received  $3000  more  than  the  daughter ;  required  the 
fortune  and  share  of  each. 

19.  A,  B,  and  C,  contemplating  the  purchase  of  a  farm, 
agreed  if  A  and  B  bought  it,  A  should  pay  |  of  the  price, 
but  if  B  and  C  bought  it,  B  should  pay  |  of  the  price, 
at  length  the  three  agreed  to  buy  it  together,  when  it 
was  found  tha  f!  p^id  $500  more  than  A ;  what  did  thr 
farm  co«t>  and  what  did  each  pay  ? 


MENTAL  ARITHMETIC. 


LESSON  VI. 

1.  Henry  is  35  years  old,  and  Mary  it  5,  in  howruanj 
years  will  Henry  be  only  6  times  as  old  LS  Mary? 

SOLUTION. — At  the  required  time,  6  times  Mary's  age  will 
equal  Henry's  age ;  then  6  times  Mary's  age,  which  is  Henry's 
age,  minus  Mary's  age,  equals  5  times  Mary's  age,  which  equals 
the  difference  between  their  ages,  which  is  35  —  5,  or  30  years, 
If  5  times  Mary's  equals  30  years,  once  her  age  equals  4.  of  30 
years,  which  is  6  years.  Hence  when  Mary  is  6  years  old  Henry 
will  be  6  times  as  old  as  she,  but  Mary  is  now  5,  therefore  in 
6  —  5,  or  1  year,  Henry  will  be  6  times  as  old  as  Mary. 

2.  James  isx28  years  old,  and  Ellen  is  8 ;  in  how  many 
years  will  James  be  3  times  as  old  as  Ellen  ? 

3.  A  is  30  years,  and  B  is  6  years  old;  in  how  many 
years  will  A  be  only  4  times  as  old  as  B  ? 

4.  Eva  is  6  years  old,  and  her  mother,  7  times  as  old; 
in  how  many  years  will  the  mother  be  5  times  as  old? 

5.  Morton  is  10  years  old,  and  Moses  30 ;  how  long 
since  Moses  was  5  times  as  old  as  Morton  ? 

6.  Jacob  is  twice  as  old  as  his  son,  who  is  20  years  cf 
age ;  how  long  since  Jacob  was  5  times  as  old  as  his  sou  t 

7.  Mary  is  |  as  old  as  her  aunt,  who  is  40  years  of 
age ;  how  long  since  Mary  was  only  ^  as  old  as  her  aunt  I 

8.  Samuel  is  J5  years  of  age,  which  is  |  of  Sarah's 
age ;  how  long  since  Sarah  was  ^  as  old  as  Samuel  ? 

9.  Jason  is  5  times  as  old  as  John,  and  the  difference 
of  their  ages  is  20  years ;  in  how  many  years  will  Jasoo 
be  3  times  as  old  as  John  ? 

10.  Henry  is  4  times  as  old  as  "William,  and  the  sum 
of  their  ages  is  25  years;  in  how  many  years  will  Henrj 
be  but  3  times  as  old  as  William  ? 

11.  |  of  A's  age  equals  |  of  B's,  and  the  difference 
between  their  ages  is  10  years;  how  long  since  A  was  8 
times  as  old  as  B? 

12 


130  MENTAL   ARITHMETIC. 

12.  A  lady  bought  some  ribbon  at  5  cents  a  yard,  and 
as  much  more  at  the  rate  of  7  cents  a  yard,  and  sold  it 
all  at  8  cents  a  yard ;  how  much  was  the  gain  on  20 
yards  ? 

.  13.  Albert  bought  some  oranges  at  2  cents  each,  and 
as  many  more  at  4  cents  each,  and  sold  them  at  the  rate 
of  2  for  8  cents,  and  gained  12  cents;  required  the  num- 
ber bought. 

14.  Witmer  bought  some  tea  at  4  shillings  a  pound, 
and  as  much  more  at  8  shillings,  and  sold  it  all  for  7 
shillings  a  pound,  and  gained  40  shillings ;  required  the 
number  of  pounds  of  each  kind. 

15.  Bowman  mixed  some  sugar  worth  5  cents  a  pound, 
with  an  equal  quantity  worth  9  cents  a  pound,  and  sold 
the  mixture  for  10  cents  a  pound,  and  gained  $6;  required 
the  quantity  of  each  kind. 


16.  Sophia  bought  a  number  of  yards  of  silk  at  the 
rate  of  3  yards  for  $1,  and  as  much  more  at  the  rate 
of  4  yards  for  $1,  and  sold  it  all  at  the  rate  of  8  yarda 
for  $3,   and  thereby  gained  $5;   how  many  yards  did 
she  buy  ? 

17.  A  boy  bought  some  apples  at  the  rate  of  4  for 
1   cent,  and  as  many  more  at  the  rate  of  5  for  1  cent, 
and  sold   them  all   at  the  rate  of  10  for  2  cents,  and 
thereby  lost  5   cents ;  how  many  of  each  kind  did  he 
buy? 

18.  \  of  M's  age  equals  |  of  N's,  and  8  years  is  f  of 
fche  difference  of  their  ages;  in  how  many  years  will  |  of 
M's  age  equal  \  of  N's  age  ? 

19.  Two  tenths  of  B's  age  equals  £  of  C's,  and  the 
sum  of  their  ages  is  30  years;  in  how  many  years  will  B 
be  3  times  as  old  as  C  ? 

20.  |  of  A's  age  added  to  f  of  B's,  which  equals  ^  of 
A's,  is  100  years;  how  long  since  A  was  5  times  as  old 
asB? 

21 .  |  of  Daniel's  age  equals  J  of  David's,  and  the  sum 


MENTAL  ARITHMETIC.  131 

of  their  ages  is  68  years ;  how  long  since  §  of  Daniel's 
age  equalled  |  of  David's  age  ? 

22.  A  boy  bought  some  apples  at  2  cents  apiece,  and 
twice  as  many  oranges  at  4  cents  apiece,  and  sold  them 
all  at  3  cents  each,  and  thereby  lost  $1 ;  how  many  of 
each  kind  did  he  buy  ? 


LESSON  VII. 

1.  A's  age  equals  3  times  B's,  but  in  10  years  A's  age 
will  be  only  twice  B's;  how  old  is  each? 

SOLUTION. — By  the  first  condition  of  the  problem,  3  times  B's 
age  equals  A's  age,  and,  in  10  years,  3  times  B's  age,  now,  -j-  10 
years,  will  equal  A's,  and  once  B's,  now,  -j-  10  years,  will  equal 
B's ;  but  A's  age  at  that  time  is  twice  B's ;  hence  2  times  (B's, 
now,  -f  10)  which  is  twice  B's  -f-  20=  3  times  B's  -f  10,  there- 
fore 3  B's,  now, — 2  B's,  now,  which  is  B's,  at  present  time, 
equals  20 —  10,  or  10  years,  and  A's  is  3  times  10,  or  30  years. 
Therefore,  &c. 

2.  John  is  5  times  as  old  as  Oliver,  but  in  8  years  he 
will  be  only  3  times  as  old ;  what  is  the  age  of  each  ? 

3.  Mary  is  ^  as  old  as  her  aunt,  but  in  20  years  she 
will  be  ^  as  old;  what  is  the  age  of  each  ? 

4.  Henry  is  1  as  old  as  his  father,  but  in  25  years  he 
will  be  |  as  old ;  required  the  age  of  each. 

5.  Ten  years  ago,  when  I  first  met  Mr.  Morgan,  I  was 
|  as  old  as  he,  but  now  I  am  ^  as  old  as  he  is ;  required 
each  of  our  ages. 

6.  Sixteen  years  ago,  when  Agnew  married,  he  was  8 
times  as  old  as  his  wife,  but  now  he  is  only  twice  as  old ; 
what  is  the  age  of  each  ? 

7.  A  hare  is  30  rods  before  a  hound,  and  runs  3  rods 
while  the  hound  runs  6;  how  many  rods  must  the  hound 
run  to  catch  the  hare  ? 

8.  Stephen  is  40  steps  before  James,  and  takes   5 
steps  to  Ja-mes's  7 ;  how  many  steps  must  James  take  to 
catch  Stephen,  supposing  their  steps  to  be  equal  ? 


132  MENTAL  ARITHMETIC. 

9.  A  "hare  takes  2  leaps  while  a  hound  takes  1,  but  1 
of  the  hound's  leaps  equals  4  of  the  hare's ;  how  much 
does  the  hound  gain  on  the  hare  in  taking  one  leap  ? 

10.  A  hare  is  30  leaps  before  a  hound,  and  takes  4 
leaps  while   the  hound   takes   2,  but  2   of  the  hound's 
equal  8  of  the  hare's ;  how  many  leaps  must  the  hound 
take  to  catch  the  hare  ? 

11.  A  fox  is  40  leaps  before  a  hound,  and  takes  3 
leaps  while  the  hound  takes  2.  but  2  of  the  hound's 
equal  4  of  the  fox's ;  in  how  many  leaps  will  the  hound 
catch  the  fox  ? 

12.  A  thief  is  20  steps  before  an  officer,  and  takes  6 
steps  while  the  officer  takes  5,  but  5  of  the  officer's  equal 
8  of  the  thief's ;  how  far  will  the  thief  run  before  he  is 
overtaken  ? 

13.  A  rabbit  is  60  leaps  before  a  hound,  and  takes  9 
leaps  while  the  hound  takes  3,  but  2  of  the  hound's 
equal  7  of  the  rabbit's;  how  many  leaps  will  the  rabbit 
take  before  being  caught? 


14.  Twenty-five  years  ago,  Willard  was  \  as  old  as  his 
uncle,  but  5  years  ago  he  was  \  as  old ;  how  old  is  each 
at  present  ? 

15.  Four  years  ago  B's  house  was  four  times  as  old  as 
his  barn,  but  2  years  hence  it  will  be  only  twice  as  old; 
how  long  has  each  been  built  ? 

16.  Three  years  ago  Emma's  doll  was  only  I-  of  the 
age  of  herself,  but  7  years  hence  it  will  be  j  of  her  age ; 
required  the  age  of  each. 

17  A  is  10  steps  before  B,  and  takes  2  steps  while  B 
takes  4,  and  4  of  A's  steps  equal  (>  of  B's;  how  many 
steps  will  each  take  before  they  are  i  ogether  ? 

18.  B  takes  30  steps  to  overtake  C;  how  far  was  C 
ahead  of  B  when  they  started,  provided  B  takes  2  steps 
while  C  takes  3,  and  2  of  B's  equal  5  of  C's  steps? 

19.  E  takes  60  steps  before  he  is  overtaken  by  D;  how 
many  steps  does  I)  take  to  catch  E,  provided  E  takes  4 


MENTAL  ARITHMETIC.  133 

steps  while  D  takes  3,  and  5  of  D's  equal  8  of  E's,  and 
how  far  ahead  was  E  when  they  started  ? 

21.  M  and  N  are  60  rods  apart  and  approach  each 
other;  how  far  will  each  travel  before  they  meet,  provid- 
ed M  takes  3  steps  while  N  takes  6,  and  2  of  M's  equal 
0  of  N's  steps  ? 

22.  A  and  B  are  150  of  B's  steps  apart,  and  approach 
each  other ;  how  many  steps  will  each  take  before  they 
are  together,  if  4  of  A's  steps  equal  8  of  B's,  and  B  takep 
9  steps  while  A  takes  3  ? 


LESSON  VIII. 

1.  What  is  the  time  of  day,  provided  ^  of  the  time 
past  midnight  equals  the  time  to  noon  ? 

SOLUTION. — By  the  condition  of  the  problem,  Jj.  of  the  time 
past  midnight  equals  the  time  to  noon,  which,  added  to  2  of  the 
time  past  midnight,  equals  J|  of  the  time  past  midnight,  which 
equals  the  time  from  midnight  to  noon,  or  12  hours.  If  3  of  the 
time  past  midnight  equals  12  hours,  1  — 1  of  12  hours,  which 
is  4,  and  ^  =  2  times  4,  or  8  hours,  the  time  past  midtight. 
Hence,  it  is  8  o'clock  in  the  morning. 

2.  What  is  the  time  of  day,  supposing  |  of  the  time 
past  midnight  equals  the  time  to  noon  ? 

3.  What  is  the  time  of  day,  if  -|  of  the  time  past  mid 
night  equals  the  time  past  noon  ? 

4.  What  is  the  time  of  day,  if  -|  of  the  time  past  mid- 
night equals  the  time  past  noon  ? 

5  What  is  the  hour  of  day,  supposing  i,  of  the  thne 
past  midnight  equals  the  time  to  midnight  again  ? 

6.  What  is  the  hour  of  day,  provided  |  of  the  dme 
past  midnight  equals  the  time  to  midnight  again? 
12* 


134  MENTAL  ARITHMETIC. 

7.  Whau  is  the  hour  of  day,  if  |  of  the  time  to  noon 
equals  the  time  past  midnight  ? 

8.  What  is  the  time  of  day,  if  U  of  the  time  to  HOOD 
equals  the  time  to  midnight  ? 

9.  A  person  being  asked  the  time  of  day,  said.  |  of 
the  time  past  noon  equals  the  time  to  midnight;  vviuit 
was  the  hour? 

10.  William  recited  his  lesson  when  f  of  the  time  part 
mon  equalled  the  time  past  midnight;  at  what  hour  did 
he  recite? 

11.  A  person  being  asked  the  time  of  day,  said,  |  of 
the  time  to  midnight  equals  the  time  past  midnight;  what 
was  the  time  ? 

12.  A  lady  being  asked  the  time  of  day,  replied,  that 
£  of  the  time  to  midnight  equalled  ^  of  the  time  past 
noon  ;  what  was  the  time  ? 

13.  Required  the  hour  of  day,  provided   ^  of  the  time 
to  midnight  equals  f  of  the  time  to  noon. 

14.  What   time,  after  12  o'clock,  are  the  hour  and 
minute  hands  of  a  watch  exactly  together  ? 

REMARK. — It  will  be  seen  that  the  minute  hand  gains  on  the 
hour  hand  11  spaces  in  going  12;  hence,  to  gain  1  space,  tho 
distance  they  are  apart  at  1  o'clock,  it  must  go  _L  of  12 
spaces,  which  is  1?  spaces,  equal  to  5JL  minutes. 

15.  What  time,   after  2  o'clock,  are   the   hour  and 
minute  hands  of  a  clock  together  ? 

16.  A  man  being  asked  the  hour  of  the  day,  replied, 
that  it  was  between  4  and  5  o'clock,  and  that  the  hour 
and  minute  hands  >vere  together;  what  was  the  time? 

17.  ^  of  the  time  past  9  o'clock,  A.  M.,  equals  |  of  the 
time  to  midnight ;  what  is  the  time  ? 

18.  A  man  being  asked  the  hour  of  day,  replied,  that 
\  of  the  time  past  3  o'clock,  equalled  £  of  the  time  to 
midnight ;  what  was  the  hour  ? 

19.  A  has  $20  in  gold  and  silver,  and  for  every  $6  of 


MENTAL  ARITHMETIC.  185 

gold  he  lias  $4  of  silver;  how  much  gold  must  be  added 
that  there  may  be  $9  of  gold  for  $3  of  silver  ? 

REMARK. — Since  the  gold  is  to  the  silver  as  6  to  4,  and  there 
are  $20  in  all,  we  find  there  are  $12  of  gold  and  $8  of  silver. 
After  the  addition,  since  3  times  the  silver  equals  the  gold,  3 
times  $8,  or  $24,  is  the  gold,  and  $24  — $12,  or  $12,  equals 
the  amount  added 

20.  A  man  has  40  pieces  of  money,  consisting  of  copper 
and  silver,  and  for  every  7  of  copper  there  are  3  of  silver ; 
how  many  pieces  of  silver  must  be  added,  that  for  everj 
•1  of  copper  there  may  be  two  of  silver  ? 

21.  A  drover  has  100  animals,  consisting  of  sheep  and 
cows,  and  he  has  2  sheep  for  every  3  cows ;  how  many 
sheep  must  he  sell  that  he  may  have  2  sheep  to  6  cows  ? 

22.  There  are  50  pupils  in  a  certain  school,  consisting 
of  girls  and  boys,  and  there  are  8  boys  to  2  girls ;  how 
many  boys  must  leave  the  school  that  there  may  be  6 
boys  to  2  girls  ? 

23.  A  man  expends  60  cents  for  an  equal  number  of 
apples  and  pears,  giving  3  cents  each  for  his  apples,  and 
2  cents  each  for  his  pears ;  how  many  pear?  must  he  sell, 
that  the  remainder  may  be  to  his  apples  as  2  to  3  i 

24.  A  man,  being  asked  the  hour  of  day,  replied,  that 
|  of\the  time  past  2  o'clock  equalled  |  of  the  time  to  mid- 
night ;  what  was  the  hour  ? 

25.  What  time,  between  7  and  8  o'clock,  are  the  hour 
and  minute  hands  of  a  watch  exactly  together  ? 

26.  In  how  many  minutes,  after  4  o'clock,  will  the 
hour  and  minute  hands  be  5  minutes  apart? 

27.  A  lady,  being  asked  the  hour  of  day,  replied,  that 
|  of  the  time  past  noon  equalled  |  of  the  time  to  midnight, 
minus  |  of  an  hour ;  what  was  the  time  ? 

'  28.  A  tree  90  feet  in  length,  by  falling,  was  broken 
into  two  parts,  such  that  |  of  the  shorter  equalled  -4  of 
the  longer;  how  much  must  be  cut  from  the  longer,  so 
ithat  ^  of  it  may  equal  |  of  the  other  part  ? 


136  MENTAL  ARITHMETIC . 

29.  A  boy  bought  24  oranges  and  lemons,  and  |  of  the 
number  of  oranges  equals  |  of  the  number  of  lemons, 
how  many  more  oranges  must  be  purchased,  that  §  oi  the 
number  of  oranges  may  equal  |  of  the  number  of  lemons? 


LESSON  IX 

1.  How  far  may  a  person  ride  in  a  coach,  which  goes 
at  the  rate  of  10  miles  an  hour,  so  that  he  may  be  gone 
but  8  hours,  provided  he  walks  home  at  the  rate  of  6 
miles  an  hour? 

SOLUTION. — If  he  walks  6  miles  in  1  hour,  to  walk  1  mile  it 
will  require  ^  of  an  hour,  and  to  walk  10  miles,  the  distance  he 
rides  in  an  hour,  it  will  require  10  times  4=  1  0;  or  5,  Of  an 
hour,  hence  5  of  the  time  he  rides  equals  the  time  he  walks, 
which  added  to  J,  the  time  he  rides,  equals  J  of  the  time  he 
rides,  which  is  8  hours,  &c. 

2.  How  far  may  a  person  ride  in  a  carriage,  going  at 
the  rate  of  9  miles  an  hour,  provided  he  is  gone  only  10 
hours,  and  walks  back  at  the  rate  of  6  miles  an  hour  ? 

3.  How  many  miles  may  I  sail  in  a  steamboat,  going 
at  the  rate  of  15  miles  an  hour,  provided  I  am  gone  only 
9  hours,  and  return  at  the  rate  of  12  miles  an  hour? 

4.  A  steamboat  whose  rate  of  sailing  in  still  water  is 
12  miles  an  hour,  descends  a  river  whose  current  is  4 
miles  an  hour,  and  is  gone  6  hours ;  how  far  did  it  go  ? 

5.  A  boat  whose  rate  of  sailing  is  10  miles  an  hour, 
moves  down  a  river  whose  current  is  2  miles  an  hour  j 
how  far  may  it  go  that  it  may  be  gone  but  5  hours  ? 

6.  Eight  men  hire  a  coach  to  ride  to  Lancaster,  but  by 
taidng  in  4  more  persons,  the  expense  of  each  is  dimi- 
nished by  $|  ;  what  do  they  pay  for  the  coach  ? 

1 .  Ten  men  hire  a  coach  for  a  certain  sum  of  money, 


MENTAL  ARITHMETIC  137 

but  taking  in  5  more  persons,  the  expense  of  each  is 
diminished  Si  ;  what  did  the  coach  cost  them  '( 

8.  Twenty  persons  engage  a  pleasure  boat  for  sailing, 
but  before  they  start,  12  of  the  company  decline  going, 
by  which  the  expense  of  each  is  increased  13 ;  what  did 
they  pay  for  the  boat  ? 

9.  A  boy  being  asked  his  age,  replied,  8  times  the 
square  of  my  age  equals  75  years ;  how  old  was  he  ? 

10.  Albert  being  asked  how  many  marbles    he   had, 
answered,  4  of  the  square  of  the  number  equals  18 ;  how 
many  marbles  had  he  ? 

11.  Three  fourths  of  the  square  of  the  number  of 
letters  in  a  sentence  equals  27;   how  many  letters  are 
there  in  the  sentence  ? 

12.  The  square  of  twice  a  number  equals  256 ;  what  is 
that  number,  and  what  is  the  square  of  ^  of  the  number? 

13.  If  |  of  the  number  of  trees  in  an   orchard  be 
squared,  the  result  will  be  100 ;  how  many  trees  are  there 
in  the  orchard  ? 

14.  The  square  of  twice  a  number  is  18  more  than 
twice  the  square  of  the  number;  what  is  the  number? 

15.  Twice  the  square  of  a  number  is  8  more  than  6 
times  the  square  of  half  the  number;  what  is  the  number? 

16.  |  of  the  square  of  a  number  is  36  more  than  |  of 
the  square  of  half  the  number ;  required  the  number. 

17.  15  is  3  more  than  |  of  the  cube  of  some  number ; 
what  is  that  number  ? 


18.  |  of  the  cube  of  a  number  is  10  more  than  the 
cube  of  |  of  the  number ;  what  is  the  number  ? 

19.  |  of  the  square  of  twice  a  number  is  equal  to  J  ol 
the  square  of  |  of  the  number,  diminished  by  3  ;  what  is 
the  number  ? 

20.  A  lady  being  asked  how  many  music  pupils  she 
had,  replied,  f  of  the  number  multiplied  by  |  of  the 
number  is  9  more  than  the  square  of  |  the  number; 
how  many  had  she  ? 


138  MENTAL   ARITHMETIC. 

21.  A  company  of  15  persons  engage  a  dinner  at  a 
hotel,  but  before  paying  the  bill  5  of  the  company  with 
draw,  by  which  each  person's  bill  was  augmented  $£• 
what  was  the  bill  ? 


LESSON  X. 

1.  In  what  proportion  must  I  mix  tea  worth  6  shilling^ 
a  pound,  with  that  worth  11  shillings  a  pound,  so  that  the 
mixture  may  be  worth  9  shillings  a  pound  ? 

SOLUTION. — If  I  take  1  Ib.  at  6.5.,  ana  sell  it  for  95. ,  I  will  gain 
9  —  6,  or  3s.,  and  to  gain  Is.  I  must  take  J  of  a  pound;  if  I 
take  1  Ib.  at  lls.,  and  sell  it  for  9s.,  I  lose  11  —  9,  or  2s.,  and 
to  lose  Is.  what  I  gained  on  the  first,  I  must  take  J  of  a  pound; 
hence  I  must  take  J  of  a  pound  at  6s.  as  often  as  J  a  pound  at 
lls.,  or,  in  whole  numbers,  6  times  J,  which  are  2  Ibs.  at  6s., 
as  of*  en  as  6  times  £,  which  are  3  Ibs.  at  11s. 

2.  In  what  proportion  must  I  mix  rice  worth  7  cents 
a  pound,  with  that  worth  12  cents,  so  that  the  mixture 
may  be  worth  9  cents  a  pound  ? 

3.  In  what  manner  shall  I  mix  sugar  worth  5  cents  a 
pound,  with  that  worth  12  cents  a  pound,  so  that  the 
mixture  may  be  worth  8  cents  a  pound  ? 

.  4.  Having  two  kinds  of  wine  worth  $2  and  89,  respec- 
tively, a  gallon,  how  shall  I  mix  them  that  the  mixture 
may  be  worth  $4  a  gallon  ? 

5.  How  many  pounds  of  sugar  worth  6  cents  a  pound 
must  I  mix  with  4  pounds  at  11  cents  a  pound,  so  that 
the  mixture  may  be  sold  for  8  cents  a  pound  ? 

6.  How  many  pounds  of  coffee  worth  8  cents  a  pound 
can  I  mix  with  12  pounds  worth  13  cents  a  pound,  sc 
that  a  pound  of  the  mixture  may  be  worth  11  cents  ? 

7.  How  many  gallons  of  wine  worth  $2  a  gallon  must 
be  mixed  with  9  gallons  at  $9  a  gallon,  so  that  the  mixture 
may  be  sold  for  $5  a  gallon? 


MENTAL  ARITHMETIC.  139 

8.  How  many  pounds  of  tea  worth  3  shillings  a  pound 
must  be  mixed  with  10  pounds  worth  9  shillings  a  pound, 
so  that  the  mixture  may  be  worth  5  shillings  a  pound  ? 

9.  A  grocer  mixed  4  pounds  of  sugar  worth  5  cents  a 
pound,  with  6  pounds  worth  10  cents  a  pound;  what  is 
the  yaJue  of  one  pound  of  the  mixture  ? 

10.  In  what  proportion  must  I  mix  sugars  worth  6,  7, 
and  11  cents  a  pound,  so  that  the  mixture  may  be  worth 
9  cents  a  pound  ? 

11.  A  merchant  mixed  5  gallons  of  wine  worth  60 
cents  per  gallon,  with  5  gallons  worth  80  cents  a  gallon ; 
required  the  value  of  a  gallon  of  the  mixture. 

12.  In  what  proportion  must  I  mix  rice  at  5,  6,  and 
12  cents  a  pound,  so  that  one  pound  of  the  mixture  may 
be  worth  8  cents  ? 

13.  How  shall  teas  worth  3,  4,  8,  and  9  shillings  per 
pound  be  mixed,  that  the  mixture   may  be  sold  for  6 
shillings  a  pound  ? 

14.  If  a  man  travels  2  miles  one  day,  4  the  next,  6  the 
next,  and  so  on,  how  many  miles  will  he  travel  the  sixth 
day? 

SOLUTION. — Since  the  distance  increases  2  miles  each  day 
after  the  first,  on  the  sixth  day,  or  5  days  after  the  first,  it  will 
increase  5  times  2,  which  are  10  miles,  which,  added  to  2  miles, 
.the  distance  he  travels  the  first  day,  equals  12  miles,  the  "distance 
he  travels  the  sixth  day. 

15.  If  a  man  travels  2  miles  the  first  day,  and  each 
day  travels  3  more  than  on  the  preceding  day,  how  far 
will  he  travel  the  tenth  day  ? 

16.  If  a  lady  pays  4  cents  for  one  yard  of  muslin,  and 
4  cents  more  for  each  yard  than  the  preceding  one,  how 
much  will  the  eighth  yard  cost  ? 

17.  If  a  person  travels  3  miles  the  first  day,  and  13  the 
sixth  day,  how  much  more  did  he  travel  each  day  than 
on  the  preceding  one  ? 

18.  How  many  days  did  a  person  travel,  supposing  he 
went  4  miles  the  first  day,  14  the  last  day,  and  went  2 
miles  more  each  day  than  on  the  preceding  one  ? 


110  MENTAL  ARITHMETIC. 

19.  If  a  person  paid  5  cents  for  one  yard  of  cloth,  and 
2  cents  more  for  each  succeeding  one,  how  many  yards 
must  be  bought  that  the  last  yard  may  cost  25  cents  ? 

20.  If  a  lady  pays  4  cents  for  the  first  yard  of  muslin, 
and  31  cents  for  the  tenth  yard,  how  much  more  did  she 
pay  for  each  yard  than  for  the  preceding  one  ? 

21.  If  Elmira  bought  12  yards  of  muslin,  paying  8 
cents  more  for  each  yard  than  for  the  preceding,  and  40 
cents  for  the  last  yard ;  how  much  did  she  pay  for  the 
first  yard  ? 

22.  A  merchant  having  sugar  worth  4,  5,  10,  and  11 
cents  a  pound,  wishes  to  make  a  mixture  of  60  Ibs.  worth 
7  cents  a  pound ;  how  many  pounds  of  each  kind  must  he 
take? 


LESSON  XI. 

1.  A  lady  bought  two  watches  and  a  chain.    The  chain 
and  gold  watch  cost  4  times  as  much  as  the  silver  watch, 
and  the  chain  and  silver  watch  cost  twice  as  much  as  the 
gold  watch.     What  is  the  value  of  each,  if  the  silver 
watch  is  worth  $30  ? 

SOLUTION. — By  the  first  condition  of  the  problem  4  times  $80. 
or  $120,  equals  the  cost  of  the  gold  watch  and  chain,  which 
added  to  $30,  the  cost  of  the  silver  watch,  is  $150,  the  cost  of 
the  two  watches  and  chain.  By  the  second  condition,  twice  the 
cost  of  the  gold  watch  equals  the  cost  of  the  silver  watch  and 
chain,  which,  added  to  the  cost  of  the  gold  watch,  equals  3  times 
the  cost  of  the  gold  watch,  which  equals  the  cost  of  all,  or  $150. 
If  3  times  the  cost  of  the  gold  watch  equals  $150,  &c 

2.  A  farmer  bought  a  horse,  colt,  and  saddle.     If  the 
liorse  be  saddled  it  will  be  worth  5  times  as  much  as  the 
colt,  but  if  the  colt  be  saddled  it  will  be  worth  ^  as  much 
as  the  horse.     What  is  the  value  of  the  horse  and  the 
saddle,  supposing  the  colt  to  be  worth  $50  ? 

3.  Tte  head  of  a  fish  is  10  inches  long;  7  times  the 


\  MENTAL  ARITHMETIC.  141 

length  of  the  head  equals  the  length  of  the  body  and  tail, 
and  3  times  the  length  of  the  tail  equals  the  length  of 
the  head  and  body.  Required  the  length  of  the  tail 
and  body  respectively. 

4.  A  person  has  two  silver  cups  and  only  one  cover  for 
both.     The  first  cup  Weighs  12  ouilces.     If  the  first  cup 
he  covered  it  will  weigh  twice  as  much  as  the  second,  but 
if  the  second  cup  be  covered  it  will  weigh  3  times  as 
much  as  the  first.     Required  the  weight  of  the  second 
cup  and  cover. 

5.  A  farmer  bought  a  cow  for  $30,  which  was  |  of 
what  he  paid  for  a  horse  and  sheep,  and  |  of  what  he 
paid  for  the  horse  and  cow  equals  what  the  sheep  cost ; 
required  the  cost  of  the  horse  and  sheep. 

6.  A  has  $20,  which  is  ^  of  what  B  and  C  have,  and 
C  has  ^  as  much  as  A  and  B  together;  how  much  have 
B  and  C  respectively  ? 

7.  A  went  to  a  store  and  borrowed  as  much  money  as 
he  had,  and  spent  4  cents ;  he  then  went  to  another  store 
arid  did  the  same,  and  then  had  4  cents  remaining;  how 
much  money  had  he  at  first  ? 

8.  A  boy  went  to  a  store,  borrowed  as  much  money  as 
he  had,  and  spent  8  cents;  he  then  went  to  a  second 
store,  borrowed  as  much  as  he  had,  and  spent  12  cents, 
and  had  no  money  remaining;  how  much  money  had  he 
at  first  ? 

9.  Rolland  went  to  a  store,  borrowed  as  much  money 
as  he  had,  and  spent  8  cents ;  he  then  went  to  a  second 
and  third  store  and  did  the  same,  and  had  no  money  re- 
maining; how  much  had  he  at  first? 

10.  A  gentleman  has  two  silver  cups  and   only  one 
cover  for  both.     The  first  cup  weighs  10  ounces,  and  if 
the  first  cup  be  covered  it  will  weigh  3  times  as  much  as 
the  second,  but  if  the  second  cup  be  covered  it  will  weigh 
5  times  as  much  as  the  first ;  required  the  weight  of  the 
second  cup  and  cover. 

11.  A  expended  10  cents  for  apples,  whicl  was  J  of 
he  expended  for  peaches  and  oranges,  and  twice 


M2  MENIAL  ARITHMETIC. 

what  he  spent  for  oranges  equals  what  he  spent  for  applea 
and  peaches ;  how  many  did  he  buy  of  each  if  the  apples 
cost  2,  the  peaches  3,  and  the  oranges  4  cents  each  ? 

12.  Reynolds  went  to  a  hotel,  borrowed  as  much  money 
as  he  had,  and  spent  2  cents;  he  then  went  to  a  second 
and  third  hotel,  did  the  same,  and  had  6  cents  remaining ; 
how  much  money  had  he  at  first  ? 

IB.  William  went  to  a  store,  borrowed  10  cents,  and 
then  spent  8  cents,  doing  the  same  at  a  second  and  third 
store,  he  found  he  had  doubled  his  money;  how  much 
money  had  he  ? 

14.  A  has  $20,  which  is  1  of  what  B  and  0  have,  and 
twice  B's  money  is  equal  to  ^  of  the  sum  of  A's  and  C's 
money ;  how  much  money  have  B  and  C  respectively  ? 

15.  James  went  to  a  store,  borrowed  10  cents,  and  then 
spent  12  cents ;  he  did  this  at  a  second  and  third  store, 
and  then  had  no  money  left ;  how  much  money  had  he 
at  first  ? 

16.  A  man  expended  $5  for  ducks,  which  was  |  of 
what  he  paid  for  geese  and  turkeys,  and  twice  what  he 
paid  for  geese  equals  what  he  paid  for  ducks  and  turkeys ; 
how  many  of  each  kind  did  he  buy,  provided   the  ducks 
cost  $|,  the  geese  $1,  and  the  turkeys  $3  each  ? 


LESSOR  XII 

1.  Two  men,  A  and  B,  in  partnership  gain  $300.  A 
owns  |  of  the  stock,  lacking  $40,  and  gains  $180;  re- 
quired the  whole  stock,  and  share  of  each. 

SOLUTION. — If  A  had  owned  5  of  the  stock,  his  gain  wotLtf 
have  been  |  of  $300,  or  $200 ;  but  he  gained  only  $180,  there- 
fore the  $40  must  gain  the  difference  between  $200  and  $180, 
which  is  $20.  If  $40  gain  $20,  to  gain  $1  it  will  require 
^0  of  $40,  which  is  $2,  and  to  gain  $300  it  will  require  300 
times  $2  which  are  $600 


MENTAL  ARITHMETIC.  143 

2.  A  and  B  anter  into  partnership  and  gam  $240.    A 
owns  |  of  the  stock,  lacking  $10,  and  his  share  of  the 
gain  is  $175  •   required  the  whole  stock,  and  share  of 
each. 

3.  Two  men  in  partnership  gain  $200.    The  first  owns 
|  of  the  stock  -f-  $40,  and  his  gain  is  $60;  what  is  the 
entire  stock  and  share  of  each  ? 

4.  A  and  B  bought  a  lottery  ticket  with  which  they 
drew  a  prize  of  $600.    A  paid  |  of  the  price  of  the  ticket, 
lacking  $12,  and  his  share  of  the  prize  was  $340 ;  what 
did  each  pay  for  the  ticket  ? 

5.  A  man  and  his  son  agree  to  mow  a  certain  field  for 
$72.     The  son  mowed  £  of  the  whole  -j-  10  acres,  and 
received  $38;  how  many  acres  did  each  mow? 

6.  Two  men  pay  $120  for  the  pasturage  of  some  cattle. 
The  first  turns  in  |  of  the  whole  number,  -(-  20,  and 
pays  $40 ;  how  many  cattle  does  each  turn  in  ? 

7.  A  and  B  agree  to  mow  a  field  of  grass  for  $60;  A 
mows  twice  as  much  as  B,  lacking  8  acres,  and  receives 
$24 ;  how  many  acres  does  each  mow  ? 

8.  Two  men  engage  to  build  a  boat  for  $84.    The  first 
labours  -l  as  many  days  as  the  second,  -f-  6  days,  and  re- 
ceives $48 ;  how  many  days  does  each  labour  ? 

9.  Two  men  receive  the  same  sum  for  labour;  but  had 
one  received  $10  more,  and  the  other  $6  less,  one  would 
have  received  5  times  as  much  as  the  other;  how  much 
does  each  receive? 

10.  A  and  B  invested  the  same  sum  in  speculation.    A 
lost  $200,  and  B  gained  $1000,  when  it  was  found  that 
B  had  4  times  as  much  money  as  A ;  what  was  the  sum 
invested  ? 

11.  Said  James  to  Isaac,  our  purses  contain  the  same 
sum  of  money,  but  if  you  give  me  $20,  and  I  give  you 
$10,  I  shall  have  3  times  as  much  as  you;  how  much 
money  had  each  ? 

12.  A  person  being  asked  the  hour  of  the  day,  replied 
that  2  hours  ago  the  time  past  noon  was  |  of  the  time  to 
midnight  2  hours  hence ;  required  the  time. 


144  MENTAL  ARITHMETIC. 

18.  Said  E  to  F  toy  age  is  5  years  more  than  yours, 
but  4  years  ago  my  age  was  ^  of  what  yours  will  be  4 
years  hence ;  what  was  the  age  of  each  ? 

14.  A  pol-3  whose  length  was  44  feet,  was  broken  into 
two  unequal  parts ;  if  the  shorter  be  increased  by  8  feet, 
and  the  longer  be  diminished  by  5  feet,  the  first  will  be  ^ 
of  the  length  of  the  second ;  required  the  length  of  each 
part. 

15.  A  staff,  whose  length  is  33  feet,  is  in  the  air  and 
water ;  and  the  length  in  the  air,  —  2  feet,  equals  4  times 
the  length  in  the  water,  -f-  6  feet ;  required  the  length 
in  the  air. 

16.  A  person  being  asked  the  time  of  day,  replied, 
that  §  of  the  time  past  midnight,  2  hours  ago,  equalled 
|  of  the  time  to  midnight,  3  hours  and  20  minutes  hence; 
required  the  time. 

17.  A  man  having  an  equal  number  of  cows  in  two 
fields,  sold  |  of  the  number  from  each,  then  7  having 
jumped  from  the  first  into  the  second,  there  were  3 
times  as  many  in  the  second  as  in  the  first ;  required  the 
number  in  each  field. 

18.  A  had  twice  as  many  oranges  as  B,  and  then  each 
losing  ^  and  A  giving  B  8,  they  each  have  the  same 
number ;  how  many  had  each  at  first  ? 

19.  C's  fortune  is  twice  D's,  but  after  each  had  spent 
|  of  his  fortune,  and  C  had  given  D  $10,  D  had  twice  as 
much  money  as  0 ;  required  the  fortune  of  each. 

20.  Two  men,  A  and  B,  agree  to  dig  a  ditch  for  850 ; 
and  |  of  what  A  digs,  increased  by  4  rods,  equals  f  of 
\vhat  B  digs,  and  B  receives  830 )  how  many  rods  did 
eaoh  dig  ? 


MENTAL  ARITHMETIC  145 


LESSON  XIII. 

1  A  lost  1 5  cents,  and  then  found  ^  as  much  as  he 
had  remaining,  and  then  had  i  as  mucn  as  he  had  at 
first,  how  much  had  he  at  first: 

SOLUTION. — After  finding  |  as  much  as  he  had  remaining,  he 
had  |  _|_  1,  which  is  |  of  what  remained  after  losing  15  cents; 
which,  by  the  last  condition  of  the  problem,  equals  ^  of  what 
he  had  at  first.  If  ^  of  what  he  had  at  first  equals  |  of  what 
remained,  g,  or  what  he  had  at  first,  equals  2  times  J,  or  |  of 
what  remained:  then  .§  of  what  remained,  which  is  what  he 
had  at  first,  minus  J>  of  what  remained,  which  is  J>  of  what 
remained,  equals  what  he  lost,  or  15  cents,  &c. 

2.  B  lost  $22,  and  then  found  i  as  much  as  he  had 
remaining,  and  then  had  ^  as  much  as  he  had  at  first; 
how  much  had  he  at  first  ? 

3.  A,  having  a  certain  sum  of  money,  found  $20,  and 
then  lost  ^  of  what  he  then  had,  and  then  had  twice  as 
much  as  he  had  at  first ;  how  much  had  he  at  first  ? 

4.  A  man  having  some  money,  borrowed  30  cents,  and 
then  losing  |  of  what  he  then  had,  found  there  remained 
3  times  as  much  as  at  first;  how  much  had  he  at  first? 

5.  A  boy  lost  44  cents,  and  then  earning  |  as  much 
as  remained,  found  he  had  |  as  much  as  at  first;  how 
much  had  he  at  first  ? 

6.  A  man  went  to  a  store  and  spent  21  cents,  and  then 
borrowing  1  of  what  he  had  remaining,  had  ^  as  much 
as  he  had  at  first;  how  much  money  had  he  at  first? 

7.  A,  at  a  game  of  chess,  won  $18,  and  then  lost  |  of 
what  he  then  had,  when,  counting  his  money,  he  found  ttt 
had  24  times  as  much  as  at  first;  how  much  did  he  make 
by  playing  ? 

8.  A  and  B  lost  12  apples,  and  then  bought  |  as  many 
as  they  lost,  and  then  had  |  as  many  as  they  had  at  first ; 
how  many  had  they  remaining? 

13* 


146  MENTAL  ARITHMETIC. 

9.  A,  in  a  game  of  chance,  lost  |  of  his  money  and 
then  won  810,  after  which  he  had  |  as  much  as  he  had 
at  first ;  how  much  did  he  lose  at  play  ? 

10.  A  and  B  lost  32  peaches,  and  then  bought  |  as 
many  as  remained,  and  then  had  j  as  many  as  at  first; 
how  many  had  each  at  first,  supposing  their  shares  to  be 
as  2  to  3? 

11.  A  boy  being  asked  his  age,  replied,  that  if  11 
years  ago  his  age  had  been  increased  by  its  j,  it  would 
then  have  been  \  of  what  it  now  is ;  required  his  age. 

12.  A  lady  being  asked  her  age,  said,  that  if  her  age 
were  increased  by  its  i,  the  sum  would  equal  3  times  her 
age  12  years  ago ;  what  was  her  age  ? 

13.  A,  B,  and  C,  dine  together;  A  furnishing  2  Ioaves3 
B  3  loaves,  and  C  contributing  25  cents  to  be  divided 
between  A  and  B ;  required  the  share  of  each. 

14.  A  boy  being  asked  his  age,  replied,  that  if  my  age 
in  3  years  be  diminished  by  its  f ,  the  remainder  will  be 
|  of  my  age  now ;  what  was  his  age  ? 

15.  A  furnished  2  loaves  for  supper,  and  B  4,  while 
C  contributed  20  cents  to  be  divided  between  A  and  B; 
how  much  of  it  should  each  receive  ? 

16.  Two  partners,  A  and  B,  lost  $210,  and  the  next 
year  gained  ^  of  what  remained,  which  was   J  of  the 
original  stock ;  what  was  the  stock  of  each;  if  |  of  A's 
equals  |  of  B's  stock  ? 

17.  A  furnished  6  eggs  for  a  repast,  and  B  10  eggs, 
while  C  contributed  16  cents  to  be  divided  between  A 
and  B;  how  much  shall  each  receive,  provided  A  and  B 
eat  the  same  number,  and  C,  4  more  than  each  ? 

18.  A  person  being  asked  his  age,  said,  that  if  my 
age  in  4  years  be  diminished  by  its  f ,  the  remainder  will 
equal  ^  of  my  age  4  years  ago ;  what  was  his  age  ? 

19.  A,  B,  and  C  eat  14  peaches,  of  which  A  owned  5, 
and  B  9,  and  C  contributed  24  cents;  how  much  of  the 
money  ought  A  and  B  each  to  receive,  if  B  eats  twice  as 
many  as  A,  and  C  eats  twice  as  many  as  B? 

20    Two  merchants   having  a  certain  number  of  yards 


MENTAL  ARITHMETIC.  147 

of  cloth,  bought  30  yards  more,  then  scld  I  as  many  as 
they  had,  and  then  had  3  times  as  many  as  at  first ;  how 
many  yards  had  each,  if  \  of  A's  number  equals  \  of  B's? 


LESSON  XIV, 

1.  A  man  bought  a  certain  number  of  sheep  for  $100, 
when  a  dog  having  killed  8  of  them,  he  sold  |  of  the 
remainder  for  cost,  and  received  $20  ]  how  many  did  he 
buy? 

SOLUTION. — If  I  of  the  remainder  cost  $20,  J  of  the  remain- 
der cost  3  times  $20,  or  $60 ;  then,  since  they  all  cost  $100, 
the  8  must  have  cost  $100— $60,  or  $40,  &c. 

2.  A  farmer  bought  a  number  of  pigs  for  $80,  when  5 
of  them  having  died,  he  sold  |  of  the  remainder  for  cost, 
and  received  $40 ;  how  many  did  he  buy  ? 

3.  A  lady  purchased  a  number  of  yards  of  muslin  for 
$1.50,  and  after  using  6  yards  she  sold  f  of  the  remain- 
der for  cost,  and  received  90  cents  less  than  it  all  cost; 
how  many  yards  did  she  purchase  ? 

4.  A  bought  a  number  of  turkeys  for  $10,  when, 
having  killed  10,  he  sold  |  of  the  remainder  for  cost, 
receiving  $8  less  than  the  cost  of  all ;  required  the  num- 
ber purchased. 

5.  A  bought  a  number  of  sheep  for  a  certain  sum,  and 
having  lost  6  he  sold  |  of  the  remainder  for  cost  and 
received  $15,  which  was  $35  less  than  they  all  sost;  how 
many  did  he  buy  ? 

6.  A  farmer  bought  a  number  of  hens  for  a  certain 
sum,  and  having  killed  10,  he  sold  |  of  the  remainder 
for  cost,  and  received  48  dimes,  which  was  72  dimes  less 
than  they  all  cost ;  how  many  did  he  retain  ? 

7.  A  bought  a  p.umber  of  ducks  for  $16,  and  having 


148  MENTAL  ARITHMETIC. 

killed  12,  he  sold  £  of  the  remainder,  lacking  8,  for 
cost,  and  received  $4;  how  many  did  he  buy? 

8.  B  bought  a  number  of  sheep  for  $30,  and  losing  2, 
he  sold  I  of  the  remainder,  lacking  3.  for  cost,  and  received 
621  less  than  all  cost;  required  the  number  bought. 

9.  Henry  bought  a  number  of  pigs  for  $48,  and  losing 
9,  he  sold  |  of  the  remainder,  minus  2,  for  cost,  receiving 
$32  less  than  all  cost;  required  the  number  purchased. 

10.  A  bought  some  calves  for  $80,  and  having  lost  10, 
he  sold  4  more  than  f  of  the  remainder  for  cost,  and 
received  $32  less  than  all  cost;   required  the  number 
purchased. 

11.  A  dog  killed  j  of  A's  sheep;  now  if  he  sells  the 
remainder  for  cost  he  will  receive  $60,  but  reserving  8 
and  selling  J  of  the  remainder  for  cost  he  will  receive 
$22 ;  how  many  had  he  at  first  ? 

12.  A  lost  §  of  his  hens,  and  found  if  he  sold  |  of  the 
remainder  for  cost  he  would  receive  40  dimes,  but  if  he 
kept  15  and  sold  |  of  the  remainder  he  would  receive  20 
dimes;  how  many  did  he  have? 

13.  B  lost  |  of  his  turkeys,  and  then  finds  by  selling 
|  of  the  remainder  for  cost,  he  would  receive  $20,  but 
finding  6,  and  selling  |  of  the  number  he  then  had,  he 
received  $24 ;  how  many  did  he  retain  ? 

14.  A  lost  |  of  his  hens;  now  if  he  finds  10  and  sells 
|  of  his  number  then  for  cost  price  he  will  receive  60 
aimes,  but  if  he  loses  10  and  sells  |  of  the  remainder  for 
cost  he  will  receive  30  dimes;  how  many  had  he  at  first? 

15.  If  60  Ibs.  of  sea  water  contain  2  Ibs.  of  salt,  how 
much  fresh  water  must  be  added  to  these  60  Ibs.,  so  that 
10  Ibs.  of  the  new  mixture  may  contain  £  of  a  pound  of 
salt  ? 

16.  Suppose  that  for  every  4  cows  a  farmer  has,  he 
should  plough  an  acre  of  land,  and  allow  one  acre  of  pas- 
ture for  every  3  cows,  how  many  cows  could  he  keep  OD 
140  acres  ? 

17.  B  lost  |  of  his  sheep;  now  if  he  finds  5,  and  sell 


MENTAL  ARITHMETIC;  149 

|  of  what  he  then  has  for  cost  price,  he  wifl  receive  $18  5 
but  if  he  loses  5,  and  sells  |  of  the  remainder  for  cost 
price,  he  will  receive  $6 ;  how  many  had  he  at  first  ? 


LESSON  XV 

1.  A  father  divided  $4400  between  his  two  childre^ 
A  and  B,  whose  ages  were  11  and  16  years  respectively, 
in  such  a  manner  that  the  parts,  at  5  per  cent.,  simple 
interest,  would  amount  to  equal  sums  when  they  became 
of  age ;  what  were  the  parts  ? 

SOLUTION. — A's  money  was  on  interest  10  years  and  B's  5 
years.  For  10  years  at  5  per  cent.,  f  of  the  principal  equals 
the  amount,  hence  |  of  A's  share  equals  A's  amount ;  and  in  the 
same  way  we  see  that  f  of  B's  share  equals  B's  amount.  Now 
the  amounts  are  to  be  equal,  hence  f  of  A's  share  =  f  of  B's 
share,  from  which  we  find  A's  share  =  f  of  B's  share ;  then  f 
of  B's  +  f  of  B's,  or  y  of  B's  share  =  $4400,  %  of  B's  — 
$400,  f  of  B's  =  $2400,  and  f  of  B's,  or  A's  =  .$2000. 

2.  A  divided  $5600  between  his  two  sons,  whose  ages 
were  11  and  15  years,  in  such  a  manner  that  the  two 
parts  on  interest,  at  five  per  cent.,  would  amount  to  equal 
sums  when  they  became  21  years  of  age;  required  the 
parts. 

3.  A   gentleman   dying  divided    $5100    among    his 
three  sons,  whose  ages  were  9,  11,  and  17  respecti\  ely, 
so  that  the  different  shares,  being  on  interest  at  5  per 
cent.,  would  amount  to  equa.  sums  when  they  became  c 
age  y  what  were  the  shares  ? 

4.  A  widow  divided  $3700  among  her  sons,  whoso 
ages  were  respectively  14,  16,  and  18  years,  in  such  a 
manner  that  the  shares,  being  put  on  interest  at  20  pei 


150  MENTAL  ARITHMETIC. 

cent.,  would  amount  to  equal  sums  when  they  became 
of  age ;  required  the  share  of  each. 

5.  A  boy  spends  i  of  his  money,-}-  $^,  then  ^  of  tne 
remainder,  -f  &z»  and  then  had  $3 ;  how  much  money 
had  he  at  first  ? 

6.  C  and  D  -together  have  20  sheep,  and  |  of  CJs 
number,  -f-  i  of  D's,  equals  £  of  C?s ;  how  many  sheep 
does  each  own  ? 

7.  A  man  spent  i  of  his  money  and  $2  more,  and  then 
spent  $2  more  than  ^  of  the  remainder,  and  then  had 
$2  remaining ;  required  his  money  at  first. 

8.  Sarah  gave  away  |  of  her  peaches,  lacking  ^  of  a 
peach,  and  then  gave  away  -|  of  the  remainder,  lacking 
|  of  a  peach,  and  then  had  5^  peaches  remaining;  how 
many  peaches  had  she  at  first  ? 

9.  A  fish  caught  in  the  Conowingo,  weighs  8  pounds, 
and  |  of  the  body,  -f-  |  of  the  head  and  tail,  weigh  as 
much  as  i  of  the  body;  required  the  weight  of  each  part, 
if  the  tail  is  |  as  heavy  as  the  head. 


10.  A  father  left  $5500  to  his  son  and  daughter,  whose 
ages  are  respectively  19  and  15  years,  so  that,  being  on 
interest  at  10  per  cent.,  the  son  should  receive  twice  as 
much  as  the  daughter,  when  they  were  21  years  of  age; 
what  was  the  share  of  each  ? 

11.  Harry  gave  ±  of  his  money,  lacking  3  cents,  to 
James,  \  of  the  remainder,  lacking  2  cents,  to  Willie, 
and  ^  of  the  remainder,  lacking  1  cent,  to  Charles,  and 
then  had  8  cents  remaining ;  what  was  Harry's  money 
before  his  gifts? 

12.  Divrde  $290  between  A  and  B,  whose  ages  arc 
respectively  15   and   19   years,  in  such  a  manner   that 
the  parts,  being  placed  on  interest,  at  10  per  cent.,  shall 
amount  to  such  sums,  at  the  time  they  are  21,  that  |  of 
A's  shall  be  equal  to  f  of  B's  money. 

13.  A  lady,  having  two  watches,  bought  a  chain  for 
$20      If  the  chain  be  put  on  the  silver  watch  their  value 


MENTAL  ARITHMETIC  151 

will  be  |  as  much  as  the  gold  watch,  but  if  it  be  pat  on 
the  gold  watch  they  will  be  worth  7  times  as  much  as 
the  silver  watch ;  what  was  the  value  of  each  watch  ? 

14.  Jordan   gave  i  of  his   money,  plus   4   cents,  to 
John,  i  of  the  remainder,  plus  3  cents,  to  George,  and 
|  of  what  now  remained,  plus  2  cents,  to  Jackson,  and 
found  he  had  4  as  much  as  at  first;  how  much  money 
had  he  at  first  ? 

15.  A  person  has  two  cups  and  a.  cover  which  weighs 
30  ounces.      If  the  first  cup  be  covered  it  will  weigh 
twice  as  much  as  the  second,  but  if  the  second  cup  be 
covered,  it  will  weigh  3  times  as  much  as  the  first ;  what 
is  the  weight  of  each  cup  ? 


LESSON  XVI. 

1.  A  farmer  bought  a  certain  number  of  sheep  for 
$60 ;  had  he  bought  5  more  at  $1  less  each,  they  would 
have  cost  him  $75;  how  many  sheep  did  he  buy? 

SOLUTION. — By  the  conditions  of  the  problem  the  5  more,  at 
$1  less  each,  cost  $75  —  $60,  or  $15,  and  one,  at  this  rate, 
cost  i  of  $15,  which  is  $3,  which,  increased  by  $1,  equals  $4, 
the  price  of  those  purchased :  hence,  there  were  as  many  pur 
chased  as  $4  is  contained  times  in  $60,  which  are  15.  There- 
fore, &e. 

2.  A  farmer  bought  a  certain   number  of  cows  for 
$200 ;  had  he  bought  2  more,  at  $2  less  each,  they  would 
have  cost  $216;  how  many  did  he  buy? 

3.  A  man  bought  a  certain  number  of  sheep  for  $80  : 
and  afterwards  bought  twice  as  many  more,  at  $2  less 
each,  and  paid  for  all  $180 ;  how  many  did  he  buy  ? 

4.  Mr.  A  bought  a  certain  number  of  turkeys  for  $5 ; 
had  he  bought  3  times  as  many,  -(-  4,  for  the  same  price, 
they  would  have  cost  him  $12  more;  how  many  did  he 
purchase  ? 


152  MENTAL  ARITHMETIC. 

5.  A  teacher  bought  a  number  of  books  for  $8 ;  had 
he  bought  4  times  this  number,  diminished  by  5,  they 
Would  have  cost  $4  more ;  how  many  did  he  buy  ? 

6.  Willis  sold  2  books  for  60  cents  each ;  on  one  he 
gained  20  per  cent.,  and  on  the  other  he  lost  40  per  cent. ; 
did  he  gain  or  lose  by  the  sale,  and  how  much  ? 

7.  What  part  of  20  per  cent,  of  6  is  25  per  cent 

of  a? 

8.  I  gave  20  per  cent,  of  my  money  to  A,  25  pei 
cent,  of  the  remainder  to  B,  and  had  $30  remaining ;  how 
much  money  had  I  at  first  ? 

9.  If  A  pays  |  of  his  salary  for  board,  what  per  cent, 
does  he  have  left  for  other  purposes  ? 

10.  If  I  sell  £  of  a  quantity  of  grain,  and  4  of  the  re- 
mainder is  spoiled,  what  per  cent,  remains  ? 

11.  Janson  sold  20  per  cent,  of  his  apples  to  A,  25 
per  cent,  of  the  remainder  to  B,  and  33-J  per  cent,  of  this 
last  remainder  to  C,  and  had  20  barrels  remaining ;  how 
many  had  he  at  first  ? 

12.  A  sold  B  a  gun  and  gained  25  per  cent.,  and  B 
sold  it  to  C  for  $24  and  gained  20  per  cent. ;  what  did 
A  pay  for  it  ? 

13.  Bought  apples  at  6  cents  for  4,  and  sold  them  at 
the  rate  of  6  for  4  cents ;  what  was  the  loss  per  cent.  ? 

14.  B  bought  melons  for  20  per  cent,  more  than  10 
cents  each,  and  sold  them  for  20  per  cent,  less  than  10 
cents  each ;  required  the  loss  per  cent.  ? 

15.  If  a  book  is  bought  for  |  of  its  value,  and  sold  for 
20  per  cent,  more  than  its  value,  what  is  the  gain  per 
cent.  ? 

16.  If  William  has  25  Der  cent,  more  money  thau 
Henry,  how  many  per  cent,  has  Henry  less  than  William? 

17.  If  a  principal  gain  A  uf  itself  in  a  year,  how  long 
will  it  require  to  double  itself? 

18.  At  25  per  cent.,  in  what  time  will  a  principal  gain 
|  of  itself  ?     |  of  itself  ?     3  times  itself? 

19.  A's  and  B's  money  on  interest  for  10  years  at  5 


MENTAL  ARITHMETIC  153 

per  cent,  amounts  to  $900 ;  how  much  Has   fcsch,  if  A's 
money  equals  3  times  B's  money? 

20.  Henry  received  $224  to  invest  in  property,  after 
retaining  12  per  cent,  on  the  amount  invested;  how  much 
did  he  invest? 

21.  If  I  gain  $20  by  selling  an  article  for  20  per  cent, 
more  than  cost,  required  the  cost  and  amount  received 

22 .  A  lost  $60  by  selling  a  horse  for  30  per  cent,  less 
than  cost ;  required  the  cost  and  amount  received. 

23.  B  bought  goods  20  per  cent,  below  par,  and  sold 
them  20  per  cent,  above  par ;  supposing  he  gained  $90, 
what  amount  of  goods  did  he  buy  ? 

24.  A  man  bought  goods  25  per  cent,  below  par,  and 
sold  them  20  per  cent,  above  par;  how  much  did  he 
invest  if  he  gained  $270  ? 

25.  A  merchant  asked  for  cloth  20  per  cent,  more  than 
cost,  but  sold  it  for  |  of  his  asking  price ;  what  was  the 
loss  per  cent.  ? 

26.  B  asked  for  flour  25  per  cent,  more  than  cost,  but 
sold  it  for  80  per  cent,  of  the  price  asked;  what  did  he 
lose  per  cent.  ? 

27.  A  grocer  asked  for  sugar  20  per  cent,  more  than 
cost,  and  sold  it  for  33 1  per  cent,  less  than  he  asked  for 
it ;  what  was  the  loss  per  cent.  ? 

28.  A  man  asked  10  per  cent,  less  for  an  article  than 
cost,  but  sold  it  for  33|  per  cent,  more  than  he  asked  for 
it ;  required  the  gain  per  cent.  ? 

29.  What  must  I  ask  for  hay,  worth  $10  a  ton,  that 
after  falling  20  per  cent.,  I  may  gain  20  per  cent,  on  the 
value  ? 

30  What  must  I  charge  for  flour,  worth  $5  a  barrel, 
that  after  falling  25  per  cent,  on  my  price,  I  may  gain  20 
per  cent,  on  the  cost? 

31.  What  must  I  ask  for  cloth,  worth  $40,  that  after 
falling  20  per  cent,  on  my  price,  I  may  gain  30  per  cent, 
on  the  cost? 

32  If  my  retail  gain  is  25  per  cent.,  and  my  whole- 
14 


154      ^  MENTAL  ARITHMETIC. 

sale  gain  is  20  per  cent,  of  my  retail  less,  what  per  cent 
*  I  gain  at  wholesale  ? 

33.  If  I  retail  at  a  gain  of  50  per  cent.,  and  sell  at 
wholesale  for  25  per  cent,  less  than  at  retail,  what  do  I 
gain  per  cent,  at  wholesale  ? 

34.  If  my  gain  at  retail  is  60  per  cent.,  and  my  gain 
at  wholesale  is  25  per  cent,  of  my  retail  gain  less,  what , 
is  my  gain  per  cent,  at  wholesale  ? 

35.  If  A's  gain  at  wholesale  is  20  per  cent.,  and  his 
gain  at  retail  is  25  per  cent,  of  his  wholesale  more,  what 
does  he  gain  per  cent,  at  retail  ? 

36.  If  B's  loss  at  wholesale  was  10  per  cent.,  and  hi? 
retail  price  was  33|  per  cent,  more ;  what  was  the  gain 
per  cent,  by  retail  ? 

37.  A  lost  40  per  cent,  of  his  flour,  and  sold  the  re- 
mainder at  a  gain  of  50  per  cent. ;  did  he  gain  or  lose, 
and  how  much  per  cent.  ? 

38.  A  barrel  of  molasses  lost  20  per  cent,  by  leakage, 
and  the  remainder  was  rold  at  a  gain  of  40  per  cent. ;  re- 
quired the  gain  per  cent. 

39.  An  article  lost  25  per  cent,  by  wastage,  and  the 
remainder  was  sold  for  20  per  cent,  above  cost;   what 
per  cent,  was  gained  or  lost  ? 

40.  A  drover  lost  33|  per  cent,  of  his  cattle,  and  sold 
the  remainder  at  a  gain  of  50  per  cent. ;  required  the 
gain  or  loss  per  cent. 

41.  The  amount  of  B's  fortune  for  4  years,  at  10  per 
cent.,  is  $200  more  than  its  amount  for  6  years  at  5  per 
cent. ;  required  the  fortune. 

42.  The  amount  of  C's  fortune  for  2  years,  at  10  per 
cent.,  is  $400  less  than  its  amount  for  6  years  at  5  per 
cent. ;  what  is  the  fortune  ? 

43.  The  amount  of  M's  money  for  5  years,  at  8  pei 
cent.,  is  $40  more  than  its  amount  for  4  years  at  6  per 
cent. ;  required  his  money. 

44.  The  amount  of  D's  money  for  2  years,  at  5  per 
cent.,  is  $60  more  than  its  interest  for  9  years  at  10  per 
cent. ;  what  is  his  money  ? 


MENTAL  ARITHMETIC.  156 

45.  The  amount  of  B's  fortune  for  5  ye&is,  at  10  per 
cent.,  is  $330  mor<«  than  the  amount  of  C's  for  the  same 
time  and  rate  per  cent.;  what  is  the  fortune  of  each, 
provided  B's  is  twice  C's? 


LESSON  XVII. 

1.  A  has  3  times  as  many  plums  as  B,  and  B  has 
twice  as  many  as  C ;  how  many  has  each,  if  A  has  12 
more  than  B  and  C  together  ? 

2.  A,  B,  and  C  can  mow  a  field  in  20  days,  A  and  B 
in  30  days,  and  B  and  C  in  40  days ;  after  the  three  had 
worked  5  days,  A  and  C  finished  it;  in  what  time  was  it 
completed  ? 

3.  A  lady  bought  10  yards  of  silk  at  the  rate  of  $4  a 
yard,  but  finding  some  of  it  damaged,  for  it  she  only 
paid  $1  a  yard,  and  thus  paid  $28 ;  how  many  yards  were 
damaged  ? 

4.  A  haviag  his  fortune  on  interest  at  5  per  cent.,  in 
one  year  spends  ^  of  his  income  in  travelling,  |  for  edu- 
cational purposes,  and  saves  $100 ;  what  is  his  fortune? 

5.  A  can  do  a  piece  of  -work  in  20  days,  B  and  C  iv 
12  days,  and  if  all  work  6  days,  C  can  complete  it  in  3 
days;  in  what  time  could  B  and  C  each  have  done  it? 

6.  A  bought  12  cows  for  $360,  and  sold  J  of  them  for 
what  all  cost ;  what  was  the  gain  per  cent.  ? 

7.  B  gained  50  per  cent,  in  each  of  3  years  on  what 
he  had  at  the  beginning  of  the  year,  and  then  had  gained 
$190  ;  what  was  his  first  capital? 

8.  A  and  B  can  do  a  piece  of  work  in  15  days,  and 
after  working  9  days  they  called  in  C,  with  whose  assist- 
ance they  finished  the  work  in  4  days;    in  what  time 
could  C  alone  have  done  it  ? 

9.  Edward  and  Ella  have  $900,  and  20  per  cent,  of 


156  MENTAL  ARITHMETIC. 

Edward's  money  equals  25  per  cent,  of  Ellas  money j 
how  much  money  has  each  ? 

10.  A  can  do  15  per  cent,  of  a  piece  of  work  in  a  day, 
B  20  per  cent.,  and  C  25  per  cent.;  after  B  had  wrought 
3  days,  and  A  2  days,  C  joined  them;  in  what  time  was 
the  work  completed? 

11.  A  man  left  $5000  to  his  wife,  son,  and  daughter; 
and  if  the  daughter  died  before  becoming  of  age,  the 
vridow  should  have  |  of  the  fortune,  but  if  the  son  died, 
she  should  have  f  of  it ;  required  the  shares  of  the  son 
and  daughter  if  the  widow  dies  ? 

12.  A  boy  bought  some  peaches  at  4  cents  each,  and 
3  times  as  many  pears  at  2  cents  each,  and  sold  them  all 
at  6  cents  each,  and  thus  gained  28  cents ;  how  many  of 
each  did  he  buy? 

13.  Three  fifths  of  A's  age  was  his  wife's  age  when 
married,  but  in  40  years,  i  of  his  age  equals  hers ;  what 
was  the  age  of  each  when  married  ? 

14.  C  and  D  ran  from  the  same  point  in  the  same 
direction,  and  when  D  had  run  40  rods,  -^  of  the  distance 
C  had  run  equalled  the  distance  he  was  ahead  of  D ;  how 
much  did  C,  in  running  40  rods,  gain  on  D  ? 

15.  What  is  the  hour  of  day  if  |  of  the  time  past  9 
A.  M.  equals  f  of  the  time  to  11  o'clock  p.  M.  ? 

16.  A  boat  whose  rate  of  sailing  is  5  miles  an  hour, 
moves  down  a  river  whose  current  is  3  miles  an  hour; 
how  far  may  it  go  that  it  may  be  back  in  10  hours  ? 

17.  A  and  B,  in  partnership,  gain  $40 ;  A  owned  | 
of  the  stock,  lacking  $12,  and  B's  share  of  the  gain  was 
$10 ;  required  the  whole  stock  and  share  of  each. 

18.  A  man  went  to  a  store  and  spent  20  cents,  and 
then  losing  |  as  much  as  remained,  had  -J  as  much  as  he 
had  at  first,  minus  $1 ;  how  much  had  he  at  first? 

19.  A  lost  20  per  cent,  of  his  peaches,  and  sold  the 
remainder  for  20  per  cent,  more  than  they  all  cost ;  what 
did  he  gain  per  cent,  on  those  sold  ? 

20.  A  drover  lost  30  per  cent,  of  his  cattle,  and  sold 


MENTAL  ARITHMETIC.  157 

the  remainder  for  10  per  cent,  more  thar  tney  all  cost  j 
what  did  he  gain  per  cent,  on  those  sold  ? 

21.  A  digs  |  of  a  ditch  in  8  days,  and  then  calling  ID 
B,  they  together  finish  it  in  9  days ;  in  what  time  could 
B  have  done  it  alone  ? 

22.  Bought  hay  for  $8  a  ton,  but  in  getting  it  I  lost 
20  per  cent,  of  it ;  what  did  it  really  cost  me  a  ton  ? 

23.  I  bought  8  apples  for  16  cents,  and  lost  |  of  them ; 
what  per  cent,  must  I  gain  on  the  remainder,  that  I  may 
neither  gain  nor  lose  by  the  transaction  ? 

•24.  A  bought  9  melons  for  36  cents,  but  losing  33^ 
per  cent,  of  them,  how  must  the  remainder  be  sold  to 
gain  33|  per  cent,  by  the  transaction  ? 

25.  The  amount  of  A's  fortune  for  3  years,  at  1.0  per 
cent.,  is  $520  more  than  the  amount  of  B's  for  5  years, 
at  6  per  cent. ;  required  the  fortune  of  each,  supposing 
A's  to  equal  3  times  B's. 

26.  If  |  of  a  barrel  of  flour  is  sold  for  what  i  of  it 
cost,  what  is  the  gain  per  cent.  ? 

27.  B  lost  20  per  cent,  of  his  maroles;  what  must  he 
gain  per  cent,  on  the  remainder,  that  he  may  gain  20  per 
cent,  on  the  whole  ? 

28.  A  barrel  of  molasses  leaked  away  20  per  cent. ; 
what  per  cent,  must  I  gain  on  the  remainder,  that  I  may 
gain  40  per  cent,  by  the  transaction  ? 

29.  A  sum  of  money  at  interest  amounts,  in  2  years, 
to  $240,  and  in  6  years  to  $320 ;  required  the  sum  and 
rate  per  cent. 

30.  A  man  sold  2  horses  for  $250 ;  on  one  he  gamed 
25  per  cent.,  on  the  other  he  lost  20  per  cent. ;  did  he 
gain  or  lose,  and  how  much,  if  he  received  for  the  second 
•|  as  much  as  for  the  first  ? 

31.  The  amount  of  a  sum  of  money  for  3  years  is  $230, 
and  the  amount  for  4  times  as  long,  at  ^  the  same  rate, 
is  $260 ;  what  are  the  sum  and  rates  per  cent.  ? 

32.  A  man  sold  2  horses  for  $210 ;  on  one  he  gained 
25  per  cent,  and  on  the  other  he  lost  25  per  cent. ;  how 

14* 


158  MENTAL  ARITHMETIC. 

much  did  he  gain,  supposing  the  second  horse  cost  him  f 
as  much  as  the  first  horse  ? 

33.  A  man  sold  a  horse  and  carriage  for  $230;  on  the 
horse  he  lost  20  per  cent.,  and  on  the  carriage  he  gained 
25  per  cent. ;  did  he  gain  or  lose,  and  how  much,  if  |  of 
what  he  paid  for  the  horse  equalled  f  of  the  cost  of  the 
carriage  ? 


NOTE. — For  additional  problems,  see  "METHODS  OF 
MENTAL  ARITHMETIC,  <fcc.,"  by  the  Author  of  this  work;  where  may 
be  found  a  large  collection,  of  an  aqausing  and  interesting  character, 
ondar  the  head  of  Social  Arithmetic. 


MENTAL  ALGEBRA. 


The  transition  from  Arithmetic  to  Algebra  being  so 
easy  and  natural,  and  a  knowledge  of  the  abbreviated 
and  general  language  of  Algebra  being  so  important,  the 
author  is  induced  to  present  the  following  short  treatise 
upon  the  subject  of  Mental  Algebra. 

METHOD  OF  TREATMENT. — The  peculiarity  of  this 
treatment  is  that  Mental  Arithmetic  and  Algebra  are 
taught  in  combination, — like  twin  sisters  going  hand  in 
hand, — one  text-book  answering  for  both  subjects.  A 
solution  is  given  to  the  first  problem  of  a  lesson,  or  class 
of  problems,  which  the  pupils  should  be  required  to 
apply  to  the  other  problems  of  that  class. 

METHOD  OF  SOLUTION. — In  Algebra  we  let  some 
symbol,  as  x,  represent  the  number  we  wish  to  find,  and 
use  this  as  we  would  the  answer  in  proving  a  problem. 
This  will  give  an  expression  or  equation  from  which 
we  may  find  the  unknown  number  represented  by  the 
symbol. 

The  following  problem  and  solution  will  indicate  the 
method  and  spirit  of  the  Algebraic  treatment.  The 
pupil  will  observe  how  much  shorter  and  simpler  it  .s 
than  the  method  of  Arithmetic. 

PROBLEM. — Henry's  number  of  apples  increased  by  3 
his 'number  equals  24;  how  many  apples  has  he? 

(159) 


160  MENTAL  ALGEBRA. 


ARITHMETICAL 
SOLUTION. 


Once  Henry's  number  plus  3  times  his  number 

equals  24. 

Hence,  4  times  Henry's  number  equals  24^, 
_  And  once  his  number  equals  £  of  24,  which  is  6 


Now  if  we  represent  the  expression  "  Henry's  number9 
by  some  symbol  as  x,  and  use  the  sign,  =,  for  the  word 
"equals"  etc.,  the  solution  will  be  much  shorter;  thus: 

ALGEBRAIC 
SOLUTION. 


NOTE.  —  In  Algebra  2x,  3x,  &c.,  mean  2  times  X,  3  times  x, 

112 
etc.,  —  j  of  x,  J  of  x,  f  of  x  are  represented  thus,  -x,  -x,  -x,  etc., 

2  t  o     o 
x   x  2x 


From  this  explanation  the  pupil  will  be  prepared  to 
understand  the  solutions  which  follow.  The  solution 
given  to  the  first  problem  of'  each  class  is  to  be  applied 
to  the  others  of  the  same  class.  The  class  is  indicated 
by  referring  to  the  page  and  problem  of  the  Mental 
Arithmetic. 

PAGE  37,  PEOB.  1. 

SOLUTION.  —  Let  x  represent  the  Let  x  =  the  number, 

number  ;  —  then  J  of  x  equals  3,  and  ^en    *  _   ^ 
x  equals  2  times  3,  or  6.     Therefore 

the  number  is  6.  and     x  =  6,  Ans. 

PAGE  58,  PROS.  1. 

SOL,  —  Let  x  equal  his  mo-  Let  x  =  his  money. 
Qey  ;—  then  x  minus  f  of  x,  or 
|  of  x  equals  60,  and  ^  of  x 

equals  J  of  60,  or  30,  and  x  an(j             ?_  _  gQ 

equals   6   times   30,    or    150.  6 

Therefore,  etc.  x   =  15°      Ans- 


MENTAL   ALGEBRA.  161 

PAGE  67,  PROB.  1. 

SOL.— Let  x  equal  the  number  of  Let  x  —  the  No. 

times ; — then,  since  the  product  of  £  _  . 

the  divisor  and  quotient  equals  the  c   3  X  *  ==  4» 

dividend,  we  have  fXz  — 4,  hence  1 

J  X  *  =  i  'rf  4,  or  2,  and  x  =  3  and     -  X  *  =  2» 

times  2,  or  6. 

x  =  6.  Ans. 

PAGE  68,  PROB.  28. 

SOL. — Let  x  =  the  number,  then,  since  the  product  of  the 
divisor  and  quotient  equals  the  dividend,  we  have  f  X  x  —  J> 
hence  J  X  x  =  f  >  and  x  =  f  •  Therefore,  etc. 

PAGE  70,  PROB.  1. 
SOL. — Let  x  equal  the  part  of  2  Let  x  =  the  part 

o 

which  equals  J ;  then  2  multiplied  2  X  *  =  -  • 

by  x  equals  J,  and  x  equals  J  of  J,  3 

which  is  |.    '  X  =  8'   An8> 

PAGE  71,  PROB.  16. 

SOL. — Let  x  equal  the  part,  then  f  times  x  equals  f ,  hence 
J  of  x  equals  f  and  x  equals  f .  Therefore,  etc. 

PAGE  90,  PROB.  1. 

SOL. — Let  x  equal  the  number          Let  x  •=  No.  of  children, 

of  children  ;  then  4z  equals  what  then,   4x  =  first  sum, 

they  received  by  the  first  condi-  and     7x  =  second  sum. 

tion,  and  7x,  what  they  received  hence, 3z—  36, 
by  the  second  condition ;    hence  x  =  12.     Ans. 

7x  —  4z,  or  3z,  equals  36,  and  x 
s=  J  of  36,  or  12.     Therefore,  etc. 

PAGE  94,  PROB.  1. 

SOL. — Let  x  =  Henry's  num-  Let  x  =  Henry's  number, 

ber ; — then  2x  will  equal  Wil-  then,    2x  =  William's  number 

liam's  number,  and  2x  plus  x,  and      Sx  =  15. 

or  3z,  equals  what  both  have,  hence,   x  =  5,  Henry's, 

or  15.    If  3z=rl5,  x= J-  of  15,  and     2z  =  10,  William's, 
which  is  5,  and  2z  =  2  times  5, 
or  10.     Therefore,  &c. 


162  MENTAL   ALGEBRA. 

PAGE  96,  PROB.  1. 

SOL.— Let  x  =  A's  number ; —  Let  x  =  A's  number, 

then  will  x  -f-  5  equal  B's  number,  then    x  -f-  5  =  B's  numoer. 

and,  adding,  2x  -\-  5  will  equal  and  2z  -f-  5  =  25. 

what  both  have,  which  is  25.     If  hence,       2x  =  20, 
2z  -}-  5  —  25,  2x  must  equal  25  x  =  10  A's. 

minus  5,  which  is  20,  &c.  x  -}-  5  =  15  B's. 

DEFINITIONS   AND   PRINCIPLES. 

Having  applied  the  above  solutions  to  the  different 
lessons,  the  pupils  will  now  be  able  to  understand  the 
following  definitions  and  principles  : — 

An  equation  is  an  expression  of  the  equality  or  two 
equal  quantities ;  thus  2x  -f-  4  =  10. 

The  quantity  on  the  left  of  the  sign  of  equality  is 
called  the  first  member,  the  quantity  on  the  right  is 
called  the  second  member  of  the  equation. 

To  transpose  a  quantity  is  to  change  it  from  one 
member  of  an  equation  to  the  other. 

We  now  present  four  simple  principles,  which  will 
enable  the  pupil  to  understand  the  following  solutions. 

PRIN.  1.  If  the  same  number  be  added  to  or  subtracted 
from  both  members  of  an  equation,  the  resulting  members 
will  be  equal. 

PRIN.  2.  If  both  members  of  an  equation  be  multiplied 
or  divided  by  the  same  number,  the  resulting  members 
will  be  equal. 

PRIN.  3.  In  transposing  a  number  from  one  member  of 
the  equation  to  the  other,  we  change  the  sign  preceding  it. 

PROOF.  Take  the  equation  3z  -f  4  =  16.  If  8x  increased 
by  4  equals  16,  it  is  evident  that  So;  =  16  —  4,  hence  in  chang- 
ing 4  from  the  first  member  to  the  second  we  change  its  sign. 
Again, 

From  the  equation  4x  =  22  —  6,  it  is  evident  that  22  is  6 
more  than  4x,  hence  4z  -{-  6  =  22,  therefore  the  6  may  be  taken 
from  the  second  member  to  the  first  by  changing  the  sign. 

PRIN.  4.  An  equation  may  be  cleared  rf  fractions  by 
multiplying  both  members  by  any  number  which  is  divi- 
sible by  all  the  denominators. 


MENTAL   ALGEBRA.  163 

PROOF.     Take  the  equation  --{--——-.    If  we  multiply  this 

2>       o        4 

by  12  we  have  6z  -f-  4x  =  30,  which  is  an  equation  without 
fractions.     Therefore,  &c. 

NOTE. — The  symbol  (  )  is  called  a  parenthesis,  and  denotes 
that  the  quantities  enclosed  in  it  are  to  be  treated  as  a  single 
quantity  ;  thus  (30  —  x)  X  3  denotes  that  30  —  z  is  to  be  mul- 
tiplied by  3,  and  =  90  —  3z,  while  30  —  x  X  3  =  30  —  3s. 


EXAMPLES    FOR   PRACTICE. 

Given,  5z  =  24  -f-  2z,  to  find  Z. 

Transposing,  5z  —  2x  —  24. 
Subtracting,  3z  =  24. 

Hence,  z  —  8.     Ans. 

2«      5 

Given,  —  --  -  =  4,  to  find  x. 

6        2i 

Multiplying  by  6,          4x  —  15  =  24. 
Transposing,  4x  =  24  -f  15  =  39. 

Hence,  x  =  9J. 

3.  Findzin3z  —  7  =  8;  In  4z  =  18  —  2z;  In  Ix  =  35  —  2x 


;  In  (8x  —  4)  X  5  =  25. 

PAGE  101,  PROS.  1. 
SOL.  —  Let  x  =  the  gain  per  cent. 
Then,  JLx  20  =  25  -20  =5. 

Hence,  -  =  5. 

o 


a;  =  25,  Ans. 


PAGE  102,  PROB.  x. 
SOL. — Let  x  =  the  cost. 
Then,  »  +  Jjc  =  25 
Or,  f»==26 

And  x  =  20      Ans. 


PAGE  104,  PROB.  1. 
SOL. — Let  a;  =  the  cost. 
Then,  Jz  =  20. 
And      zr=80.    Ans. 


MENTAL   ALGEBRA. 

PAGE  118,  PROS.  1. 

SOL.—  Let  x  =  what  they  pay  for  .me  eow 
Ihen,  4x  =  what  A  pays. 

And  5x  =  what  B  pays. 

Then,4*-f  5z:=36. 
Or,  9ar  =  36, 

x~4. 

4z=16,  A's  part. 
bx  '  =  20,  B'spart. 

PAGE  120,  PROB.  1. 
SOL.  —  Let  4z  =  A's  share. 
And  6z  =  B's  share. 

Then,  4z  -}-  6z  :=  30, 
Or,  lOa:  —  30. 

z  =  3. 

4z=12,  A's  share. 
6a;  =  18,  B's  share. 

PAGE  122,  PROS.  9. 

SOL.  —  Let  x  =  the  time  in  which  both  can  eat  it. 
Then,          .-  =  what  both  can  eat  in  one  day. 

|  =  what  Fuller  can  eat  in  a  day. 
—  =  what  Brodhead  can  eat  in  a  day 


5x  =  12. 
2  =  2§.     Ans. 

PAGE  126,  PROS.  1. 

SOL.  —  Let  z  =  number  of  idle  days. 
Then  30  —  x  =  number  of  working  days 

(30  —  x)  X  3  =  what  he  earned. 

x  =  what  he  forfeited. 
Hence,         6C  -f  x  =  (30  —  x)  V  3. 
Or  60  -fz=:90  —  8z. 

4x  =  40. 
x  =  10,  Ans. 


MENTAL  ALGEBRA. 


165 


+  8  =  24,  the  tail. 


PAGE  127,  PROB.  8. 
OL. — Let  x  =  length  of  body. 

Then,  |  +  8  =  length  of  tail. 

And  8  =  length  of  head. 

Hence,  2x  =  16  -f-  — 

And  -  =  16. 

x  =  32,  the  body. 
Then, 

32  -f  24  +  8  ==  64,  Ans. 

PAGE  129,  PROB.  1. 
SOL. — Let  x  =  the  time. 
Then,  35  ^-j-  x  =  H's  age  at  that  time. 

And  5 '-}-  x  =  M's  age  at  that  time. 

Hence,  (5  +  x)  X  6  =  35  -f  x. 
Or,  30  -{-  6*  —  35  -f-  x. 

Hence,  5z  =  5. 

z  =  l,  Ans. 

PAGE  130,  PROB.  16. 
SOL. — Let  x  =  the  number  of  each  kind. 
Then,  2x  =  cost  of  first  kind, 
And     4x  =  cost  of  second  kind. 
And     Qx  =  cost  of  both  kinds. 

Since  there  were  2x  in  all  and  they  were  sold  at  the  rate 
of  4  cts.  each, 

8x  =  what  they  were  sold  for. 
Then  Sx  —  6x  =  12, 
2x  =  12. 
x  =  6,  Ans. 

PAGE  131,  PROB.  1. 
SOL.— 
Let  z  =  B's  age. 


3x  =  A's  age. 

x  -\- 10  3E=  B's  age  in  10  years. 

Sx  -j-  10  =  A's  age  in  10  years. 


f  10  =  2x  -f20. 
+  10  =  20. 

x  =a  10,  B's  age. 


16 


PAGE  133,  PROB.  1. 
SOL.— 
Let  x  =  time  past  midnight. 

x 

-  =  time  to  noon. 

A 

X 

x  -f-  ^  =  12  hours. 


MENTAL   ALGEBRA. 

PAGE  134,  PROB.  15. 
SOL.—  Let  x  =  distance  the  minute  hand  goes. 

x 
Then,      —  =  distance  the  hour  hand  goes. 

Andz -  — 10   the   number  of   minute   spaces   they  are 

12  ~~      '  apart  at  2  o'clock. 


Ilz=rl20. 

z  =  10{£,  Ans. 


PAGE  136,  PROS.  1. 
SOL.  —  Let  x  =  the  distance. 


Then,          —  =  time  in  going. 


And  -  =  time  in  returning. 


— 
- 


And    6z-f  3z=240. 
Sx  =  240. 
z  =  30,  Ana. 

PAGE  136,  PROS.  6. 

SOL.—  Let  x  =  what  they  pay  for  the  coach 
Then,          -  =  what  each  of  eight  pay. 

And  —  =  what  each  of  twelve  pay. 


3*  —  2z  =  18. 

«  =  18  Ans. 

PAGE  140,  PROB.  1. 
SOL.—  Let  O  =  value  of  the  gold 

C  =  value  of  the  chain. 
Then,  2G  =  C  +30. 

And  a+  C  =  30X4  =  120. 

Or,  67  =  120—  (7. 

Putting  this  in  the  1st,         2G  =  120  —  G  +  30. 
Transposing,  3  O  =  150. 

G  =  50  Ans. 

(7^120  —  50  =  70  Ans. 


MENTAL   ALGEBRA.  167 

PAGE  145,  PROS.  1. 

SOL.  —  Let  x  =  what  he  had  at  first 
Then,       x  —  15  =  sum  after  losing  15  cents, 
And     |(z  —  15)  =  sum  after  finding  J  of  remainder 

Then,  |(*-16)  =  g. 

Or,          f*-20  =  |. 
Hence,  fa;  =  20. 

*=±r4 
6 

3  =  24  Ans. 

PAGE  150,  PROS.  14. 

SOL.  —  Let  S—ib.Q  value  of  the  silver  watch, 
G=  the  value  of  the  gold  watch, 
0  =  the  value  of  the  chain. 
Then,    G=ZS+3C. 
And     7S=G+C.     Putting  for  G  its  value  we  have 

7S=3S+ZC+  C. 
Or,      4S=4C. 

S  =  (7=:  20,  Ans. 

G=  40X3  =  120,  Ans. 

The  following  equations  are  a  brief  statement  of  the 
rules  for  all  the  cases  of  Simple  Interest.  The  rate  is 
expressed  as  a  fraction;  thus,  5  per  cent,  is  used  as 
Tfo  or  .05. 

Let  p  =  principal.    (1)  i  =p  X  r  X  t.  i 

i  =  interest.  i  \*J  T 

'  ' 


A  =  amount.      (3)  t  = 


EXPLANATION.  —  The  1st  denotes  that  the  interest  equals  the 
product  of  the  principal,  rate,  and  time.  Thus  the  interest  of 
$40  for  5  yrs.  at  6  per  cent,  equals  $40  X  lib  X  5  =  $12 
The  2d  denotes  that  the  principal  equals  the  interest  divided  by 
the  product  of  the  rate  and  time,  etc. 


168  MENTAL   ALGEBRA. 

To  TEACHERS. — The  author  presents  this  little  treatise 
upon  Mental  Algebra  to  his  friends  who  have  exhibited 
such  a  kind  appreciation  of  his  former  labors,  with  the 
following  remarks  and  suggestions  : — 

The  plan  of  combining  Mental  Arithmetic  and  Mental 
Algebra  in  one  book,  is,  so  far  as  the  author  knows, 
a  new  one,  and  is  therefore  a  distinctive  feature  of  this 
work.  Its  advantages  are  supposed  to  be  important,  a 
few  of  which  will  be  briefly  stated. 

First,  There  is  an  economy  in  time  and  money ; — 
secondly,  there  is  greater  convenience  in  teaching; — 
thirdly,  it  is  in  accordance  with  the  logical  relation  of 
the  subjects; — fourthly,  the  arithmetical  solution  will 
assist  in  understanding  the  algebraic,  and  the  algebraic 
will  often  throw  light  upon  the  arithmetical. 

METHODS  OF  TEACHING.  For  those  who  desire  to 
try  the  plan  here  presented,  developing  the  twin  sisters 
together,  we  present  the  following  methods  of  instruc 
tion  : — 

1st.  The  same  problem  may  be  solved  first  arith- 
metically and  then  algebraically,  by  the  same  or  different 
pupils. 

2d.  The  algebraic  solution  may  be  deferred  until 
several  or  all  of  a  class  of  problems  have  been  solved 
arithmetically. 

3d.  The  book  may  first  be  completed  arithmetically 
and  then  reviewed  algebraically.  It  is  probable  that 
the  second  method  will  be  preferred  by  most  teachers. 

If  the  pupils  are  carefully  drilled  upon  the  problems 
under  the  several  classes  which  we  have  solved,  they 
will  be  able  to  apply  the  algebraic  method  to  other 
problems  not  included  in  these  classes. 


THE    END. 


OVERDUE. 


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